Tree of quantum_mechanics2

The Tree of Quantum Mechanics
The Tree of
Quantum
Mechanics
by
J.THAMBAJI
The Roots
Electromagnetic Theory 1860
Scientific Method ~1614
The Trunk
Hydrogen Atom Model 1913
Blackbody Studies 1859
Branches & Leaves
Theory of Everything
Electrons are Waves 1923

The Tree of Quantum Mechanics - Overview
Often, at this museum, I have used our Spectroscopy Cart to explain to the public
how we know what the sun and stars are made of. The explanation involves
showing the spectra of a few elements; and explaining that the lines of each are
different; so we can tell them all apart. Occasionally, some brash kid (and a few
adults) might ask, “why are there lines?”. When I replied that the answer lies
within quantum mechanics, the bulk of my audience would plead urgent business
somewhere else, and melt away. It’s as if those 2 words somehow caused their
critical faculties to shut down, much as “calculus” did a generation or 2 back. To
those few who remained, I would offer a historical summary of the main ideas,
and those who advanced them. Many thanked me profusely for so opening
their eyes.

As most of today’s physics, chemistry, and molecular biology depends on
quantum mechanics, as well as many of our gaudiest consumer goods, I felt that
a Museum of Nature and Science should attempt to convey the core ideas to the
public. The problem is that the field is essentially mathematical; and only a few
of our museum staff have the necessary training. This presentation grew out of
many discussions between us on what to present, and how. Only you can judge
how well we have succeeded.
What you will see is in the form of an historical tree, with roots stretching back to
Galileo, a trunk embodying the development of the main ideas, and branches
and leaves containing modern developments and possible future trends.

Expect no mathematics. On the Tree Menu each box with red text cites a critical
development or insight; and is identified by a number on the bottom right. Boxes
with green text lead to explanatory and biographical material. The numbers are
in a rough historical sequence.
Within the text slides you will see arrows to navigate forward through the slides
for each topic. To return to the main “Tree” menu click on the tree icon on the
bottom right of every slide. Red and green text will link to their corresponding
slides in the presentation.
We recommend a quick forward stroll through the red boxes, followed by
meandering through those green details that pique your interest.
Dave Sonnabend and the Museum Staff

Scientific Method
~1614 1
Particle theory of light
1666 2
Conservation
of energy 5
Thermodynamics,
Statistical mechanics
1824 6
Wave theory of Light
1678 3
Interference
1801 4
Electromagnetic Theory
1860 7
Huyghens
Young
Faraday
Who of Thermodynamics
Newton Laws of Physics
Galileo
Blackbody Studies 8
Ultraviolet Catastrophe
~1880 10
Discrete Energy Levels
1900 12
Photoelectric Effect
1905 13
Hydrogen Atom Model
1913 15
Spectroscopy
1859 9
Discovery of Electrons
1897 11
Discovery of Nucleus
1913 14
Bunsen
Thomson
Rutherford
Kirchhoff
Rayleigh
Planck
Einstein
Bohr
Electrons are waves
1923 16
Matrix, Wave Mechanics
1925 17
Statistical Interpretation
~1926 19
Quantum Electrodynamics 21
Quantum Chromodynamics 27
Electron Diffraction
1927 18
Modern Chemistry, Molecular Biology 22
Solid State Theory: Semiconductors,
Superconductors 23
Astrophysics 24
Weirdness 25
Theory of Everything 28
Periodic
Table
Gell-Mann,
Ne’eman
De Broglie
Heisenberg
Born
Feynman, et. al.
Special Relativity
1920 20
Nuclear Physics 26
Nuclear Medicine,
Weapons,
Cyclotrons
Click on text
to select
Who of Statistical Mechanics
Maxwell
Newton
Schrodinger
Pauli
Dirac
Click on text
to select

If this were a history of science, we would begin millennia earlier, and fill many
more pages. However, as quantum mechanics is our theme, we have chosen to
begin our story with the intellectual revolution started by Galileo. The road leads
through the birth of physics with Newton, the first tentative steps to
understanding how the world ticks in the 17th and 18th centuries, and the major
leaps forward in the 19th. The direct path to quantum mechanics began with the
researches of Gustav Kirchhoff; and if impatience rules your soul, you may skip
directly to 8 Blackbody Studies. Else, we recommend a more leisurely stroll
through the main growth of physics.
1: The Scientific Method
- The Roots

What is today called “The Scientific Method” was first introduced by Galileo. In
his several writings, beginning around 1610, he argued that no theory should be
accepted without supporting observational evidence. Put differently, if your
measurements or observations conflict with your theory, you should modify or
reject the theory. You might then hope to find a better theory, consistent with all
known measurements and observations. This intellectual philosophy largely
arose from his telescopic astronomical observations, which strongly contradicted
the thinking of Aristotle (384 BC – 322 BC), as expounded in the “Almagest” of
Claudius Ptolemy (~100 – ~170), around 150 AD.
- The Roots
Portrait of Galileo by Giusto Sustermans,
1636

He found 4 satellites of Jupiter, which plainly didn’t go around the earth. He saw
that Venus had phases like the Moon, coinciding with its apparent path near the
sun – it clearly orbited the sun. He found mountains on the moon, showing that it
wasn’t a “perfect” body. Finally, the sun has spots which come and go, contrary to
Aristotle’s view that the heavens are unchanging. All this was a big problem for the
Catholic Church, which had slowly adopted Aristotle’s views as dogma, following
the persuasive writings of Saint Thomas Aquinas (1224/5 – 1274) in his “Summa
Theologica” published over the years 1265 – 1273.
- The Roots
Observations of Jupiter's moons
Phases of Venus

The Tree of Quantum Mechanics - The Roots
Galileo’s thinking was quickly adopted by the intellectual community in Europe, in
large part because they were mostly out of the reach of the Catholic Church. While
much valuable work had come before, the real flowering of science began here.
That the scientific method could spread so easily was, in part, due to the
Reformation, begun in 1517, with the publication of the writings of Martin Luther
(1483 – 1546).
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Dialogo di Galileo GalileiGalileo discovering the
principle of the pendulum

1: Galileo
Galileo Galilei (1564 – 1642) was born in Pisa, Italia. His early education was at the
Monastery of Santa Maria at Vallombrosa, and then at the University of Pisa in
medicine in 1581. Lacking interest in this, He left without a degree in 1585, to study
privately in philosophy and mathematics. His first discovery was that the period of
a pendulum is independent of its amplitude, if it doesn’t swing too far. This finding
was to dominate clock design for centuries thereafter. In 1589 he was appointed
professor of mathematics at the University of Pisa. His researches there on falling
bodies showed that Aristotle’s belief that heavier bodies fall faster was incorrect;
but the tale that he demonstrated this by dropping weights from the leaning tower
of Pisa isn’t well substantiated. By challenging Aristotle in this and other ways he
earned the ire of most of the faculty; and in 1592 he wasn’t reappointed.
He left to take up the chair of mathematics at the University of Padua, where
he remained till 1610.

1: Galileo
The Copernican heliocentric theory had been published in 1543 (De revolutionibus
orbium celestium, banned as heretical by the Catholic Church); and by 1595 Galileo
was leaning toward it. However, it was the supernova of 1604 that really sparked
his interest in astronomy. The invention of the telescope is generally attributed to
Hans Lipperhey (1587 – 1619), a Dutch lens maker, in 1608. When Galileo heard of
this in the next year, he built his own device, and turned it to the heavens. He was
the first to see 4 moons of Jupiter, the rings of Saturn, mountains on the moon,
sunspots, and the phases of Venus – all generally incompatible with the teachings
of Aristotle (384 – 322 bc) and Claudius Ptolemy (~100 – ~170). He published these
findings in The Starry Messenger, which became immensely popular throughout
Europe.
Cover page from Sidereus Nuncius
- The Roots

1: Galileo
While most intellectuals sided with Galileo, some powerful enemies caused the
matter to be referred to the Inquisition. This body held that, as Church doctrine
had accepted Aristotle’s teaching, the telescopic evidence was irrelevant, and
convicted Galileo of heresy. In 1633 they sentenced him to life imprisonment, more
or less enforced until his death. During that period, he wrote Dialogue Concerning
the Two Chief World Systems, in which his 3 characters argued for and against
heliocentrism, without taking an explicit position. However, Pope Urban VIII saw
Simplicio, Galileo’s defender of Aristotle, as himself. The book was banned, and all
copies ordered burned. It was smuggled out of Italy, and published in Holland in
1638. As a postscript, the Church formally apologized to Galileo in 1992. While he
made several important inventions, and contributed greatly to mechanics, the
scientific method and his astronomical discoveries were to remake the world.
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Galileo facing the Roman Inquistion
by Cristiano Banti (1857).

2: Light is Particles
What is light? Many ancients attributed light to particles moving at some extreme
speed. But whether these particles originated in the eye, or in the object viewed,
was unresolved. In 1666, Isaac Newton allowed sunlight to pass through a small
hole in a blind, and fall on a glass prism. It was bent (refracted) there, and again at
a second face of the prism. It then fell on a screen, and showed a rainbow of colors
(see figure). The setup may be regarded as the first spectroscope. This separation
of white light into its many colors is called “dispersion”. Newton thought that light
consisted of particles; though why “blue” should bend more in the prism than
“green”, and these more than “red”, he couldn’t say. Another difficulty lacking
explanation was why intersecting light beams didn’t interact. After all, 2 beams of
particles ought to bounce off each other. Not until the 20th century, would these
issues be clarified, and in such a way as would have astounded even such an
intellect as Newton.
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2: The Laws of Physics
That the science of mechanics could be founded on an axiomatic basis was the
central accomplishment of Isaac Newton’s illustrious career (see 2: Newton). Part
of his inspiration was “Kepler’s Laws” of planetary motion. Johannes Kepler (1571 –
1630), by analyzing the planetary observations of Tycho Brahe (1546 – 1601),
certainly the most accurate of the time, determined that elliptical orbits best fit the
data. He eventually published his findings as 3 laws:
1. Each planet moves in an ellipse, with the sun at one focus.
2. The radius vector of each planet sweeps out equal areas in equal times.
3. The square of the period of revolution of each planet is proportional to the
cube of the mean distance of the planet from the sun.
Johannes Kepler , 1610

Unlike Newton’s laws to follow, these were purely phenomenological – i.e., they
were a straight generalization of Tycho’s observations. As these observations were
of the apparent paths of the planets against the background stars as seen from a
moving earth, Kepler’s insight was extraordinary.
In his early days at Trinity College, Newton collected the known work on mechanics,
in large part the writings of Galileo (see 1 Galileo), and formulated his 3 laws of
motion. In modern language these are:
1. Every body continues in its state of rest or of uniform motion in a straight line,
except in so far as it is compelled by forces to change that state.
2.Change of motion is proportional to the force and takes place in the direction
of the straight line in which the force acts.
3.To every action there is always an equal and contrary reaction; or, the
mutual actions of any two bodies are always equal and oppositely directed
along the same straight line.
- The Roots

In addition, his law of gravitation may be stated as, “Between any 2 bodies, there is
an attractive force proportional to the product of their masses, and inversely
proportional to the square of the distance between them”. Newton published all
this in his monumental Philosophiae Naturalis Principia Mathematica, arguably, the
most important scientific treatise of all time. Where did he get it? In later life, he
said that it came to him while watching an apple fall from a tree. In the book, he
used a long set of geometrical arguments based on Kepler’s Laws. However, it’s
more likely that he used his own method of fluxions (calculus) to prove it in a few
steps from Kepler’s Laws.
Sir Isaac Newton's own first edition copy of
his Philosophiae Naturalis Principia
Mathematica. The book can be seen in the
Wren Library of Trinity College, Cambridge.

Later, using only his laws of mechanics and gravitation, Newton proved Kepler’s
laws. If this set of advances seems circular, keep in mind that Kepler’s insight was
purely phenomenological, i.e., it was based solely on Tycho’s observations.
Anyway, the science of physics was firmly underway, and may be said to have
begun with the publication of the Principia in 1687. As a postscript, Henry
Cavendish (1731 – 1810), in 1798, in an elegant laboratory experiment, managed to
measure the constant of proportionality G in Newton’s law of gravitation, thus
determining the mass of the sun, earth, and those planets with moons.
As we’ll later see, “force” is a concept of classical physics; quantum mechanics
views motion quite differently.
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Drawing of torsion balance apparatus used by Henry
Cavendish in the 'Cavendish Experiment' to measure
the gravitational constant in 1798. This is a vertical
section through the apparatus, including the building
that housed it.

2: Newton
Isaac Newton (1642 – 1727) was born at Woolsthorpe Manor, some 100 km
northwest of Cambridge. After a rather unhappy, fatherless childhood, his
intellectual prowess was noted; and his mother, reluctantly, allowed him to further
his education. In 1661, he entered Trinity College, Cambridge, as a “subsizar”, the
bottom of their totem pole. He got free board and tuition, in exchange for menial
service. In the next year, his first exposure to 17th century mathematics began. In
his notebooks, he speculated on many subjects. Perhaps the most forward looking
was his geometrical proof of what today is called “the fundamental theorem of the
calculus” – essentially that derivatives and integrals, that is, slopes of a curve, and
the area under that curve, are inverses of each other; a truly remarkable insight, on
which he would later expand.
Newton in a 1702 portrait by
Godfrey Kneller

The years 1664 – 1666 were perhaps the most productive, when he
formulated the laws of mechanics and gravitation (see 2: Laws of
Physics). In 1665 he graduated with a Bachelor of Arts degree.
Also in that year, the bubonic plague reached Cambridge; the
University was closed; and Newton returned to Woolsthorpe.
There, he performed a number of experiments on the refraction
of sunlight in a prism (see 2 Particle Theory of Light).
Back in Cambridge, he was elected a minor fellow of Trinity College, which allowed
him to remain. A year later he received a Master of Arts, and was elected a major
fellow. In 1668, Newton’s professor, Isaac Barrow retired as the first Lucasian Chair
of Mathematics, to join the ministry, and recommended Newton to succeed
him. Newton held this chair for nearly 30 years.
- The Roots
2: Newton

2: Newton
Just prior to this appointment, Newton designed and built the first reflecting
telescope. For the same resolution and light gathering power, it’s much shorter
than the Galilean equivalent, which is built entirely of lenses. Lens based
instruments suffer from chromatic aberration – variation of focal length with
wavelength or color. Today we know that the speed of light in glass, and thus
refraction, varies with wavelength. The effect is partly corrected in achromats,
doublet lenses made of 2 different kinds of glass, or better (and more expensive)
anachromats (triplets). Mirrors aren’t subject to chromatic aberration (see our
Telescope Exhibit).
Eyepiece
Primary
Mirror
Diagonal
Mirror
Newtonian Telescope

2: Newton
Today, most amateur telescopes are based on Newton’s design. Together with
other advantages, telescopes with large mirrors have entirely supplanted lens
based designs in modern astronomical instruments. In 1671, Barrow took Newton’s
telescope to the Royal Society in London. It was so admired, that Newton was
promptly elected a fellow. Much later, in 1703, Newton was elected President of
the Society, which he ruled with an iron hand.
A replica of Isaac Newton's telescope of
1672

2: Newton
Newton’s advances in mathematics began about this time, including original work
on infinite series, and the invention of the binomial theorem and “the method of
fluxions”, later called “calculus”. He had started by devouring the works of René
Descartes (1596 – 1650), Blaise Pascal (1623 – 1662), and others. Descartes had
pioneered coordinate (analytic) geometry, although Newton’s taste later drifted
back to the Greek view of the subject. Much later, Karl Friedrich Gauss (1777 –
1855), “The Prince of Mathematicians” referred to Newton as “The First Geometer
of His Age”, high praise indeed.
Newton in a 1702 portrait by Godfrey Kneller

2: Newton
Exposure to the larger scientific community brought out the contentious side of
Newton – temper; he couldn’t take criticism. Robert Hooke (1635 – 1703) sent him
up the wall by disagreeing with his ideas on optics. Later, it went from bad to
worse, when Hooke claimed that Newton had stolen some of his ideas, causing
Newton to have a complete nervous breakdown. Somewhere around 1675, both
Newton and Gottfried Wilhelm Leibnitz (1646 – 1716) developed the differential
and integral calculus, leading to further controversy over who was first. Charges of
plagiarism went back and forth for several years, adding to Newton’s paranoia.
Scholars today are generally agreed that Newton was first; but Leibnitz was first to
publish.
Isaac Newton, 1689
Gottfried Wilhelm Leibnitz,
1700

Of Newton’s many achievements, today we would regard the publication of
Philosophiae Naturalis Principia Mathematica in 1687 as number one. This work
covered his studies in optics, laid an axiomatic foundation for the science of physics,
introduced the law of gravitation, and derived Kepler’s laws of planetary motion
(see 2 Laws of Physics). Newton had been quite reluctant to publish his many notes
covering this material, but was strongly urged to do so by the astronomer Edmond
Halley (1656 – 1742), who corrected the proofs, and saw to its publication.
Incidentally, Halley didn’t discover the comet named for him; but he computed its
orbit, and accurately predicted its return in 1758. He also showed how to size the
solar system through multiple observations of transits of Mercury and Venus across
the sun. The method was applied to transits of Venus in 1761 and 1769.
- The Roots
2: Newton
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In 1678, Christian Huyghens (1629 – 1695) (see 3 Huyghens) advanced the idea that
light consisted of waves. He argued that light traveled slower in glass than in air,
with blue light slower than red. This concept of refraction is illustrated in the figure,
using the marching men analogy. Rows of soldiers replace the wave fronts of light.
At the top, they march on an easy road (air); while at the bottom they slog more
slowly through mud (glass). Short soldiers can keep up on the left by taking shorter
steps than tall; but have more difficulty in the mud, go slower, and thus diverge
from their faster brethren. The distance between wave fronts is the wavelength,
shorter in glass and for short soldiers.
- The Roots
3. Light is Waves

Waves avoid the corpuscle problem of crossing beams. After all, water waves pass
through each other without interaction. On the other hand, waves of what? Light
reaches us from the sun and stars; so, presumably, there must be something in
space to wave. Following Huyghens (and the Greeks), this stuff was called the
“ether”, or more pompously, “the luminiferous aether”. Like sound waves, and
surface waves in water, light was assumed to be very rapid displacements of the
ether. The conflict between waves and particles persisted until 1801 (see 4
Interference), when waves won out. However, a century later, equally compelling
evidence favored particles (see 13 Photoelectric Effect); and we had to face the
mind wrenching notion that light is both.
- The Roots
3. Light is Waves
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Christian Huyghens (1629 – 1695), was a Dutch polymath. He was born in Holland,
but spent much of his life in France, even during its war with Holland (1672 – 1678).
One of his first achievements was the design and construction of the first truly
accurate pendulum clock. Galileo had shown that a pendulum had a quite stable
period, provided the amplitude was small (see 1 Galileo). Huyghens showed that by
restraining the upper part of the suspending wire by a pair of cycloidal surfaces, the
period would be stable over much wider swings. This was analyzed and described
in his Horologium Oscillatorium in 1656.
- The Roots
3: Huyghens
Huyghens in a 1671 portrait by Caspar Netscher

His astronomical contributions began by finding an improved method of grinding
lenses, somewhere around 1654. He used this to build an improved telescope, with
which he discovered Titan, the largest moon of Saturn in 1655; and in 1659, he
showed that Saturn’s rings are truly circular. As a postscript, the European
spacecraft Huyghens was carried to Saturn by the NASA spacecraft Cassini, where it
landed on Titan, greatly improving our knowledge of that body.
Huyghens is probably best known for his studies of light, which, unlike Newton, he
regarded as made up of waves. In this view, each point on an expanding wave front
is the source of a spherically expanding wavelet; the envelope of the wavelets is the
wave front. Following the discovery of interference (see 4 Interference) by Thomas
Young (see 4 Young) in 1801, this view was to dominate physics until quantum
mechanics came along a century later.
- The Roots
3: Huyghens
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4: Interference
In 1801, Thomas Young (1773 – 1829) (see 4 Young) performed an experiment that
essentially demolished Newton’s corpuscular theory of light. In his original setup,
he allowed sunlight to fall on a screen with a pinhole. Light passing through the
pinhole then impinged on a second screen with 2 pinholes. Finally, light from the
second screen struck a third screen, where a complicated set of colored fringes
appeared. This made no sense from the corpuscular viewpoint.

4: Interference
What was going on may be seen more clearly from a later modification of Young’s
experiment. As shown in the figure (not to scale), light from a source L goes
through a filter F, passing only a single color. It then strikes a screen S with a single
slit, running out of the plane of the figure. Light from S then lands on a second
screen T with 2 slits parallel to the slit in S. Light from these 2 slits lands on a
final screen P, where one sees a wavy pattern of intensity, as shown. That this
is due to interference is shown by covering one of the slits in T, when the wavy
final pattern on P is replaced by a single intensity peak.

Measurements on P show that at any peak such as C, the distances to the 2 slits in T
differ by an integral number of wavelengths of the color passed by F; while at a null
in intensity such as D, the difference in distance is an odd number of half
wavelengths. Experiments such as this “established” the wave nature of light, and
also pinned down the wavelength associated with each color.
All this was turned on its head with the arrival of quantum mechanics, early in the
20th century. For a modern reinterpretation of the Young experiment, see 19
Statistical Interpretation.
- The Roots
4: Interference
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4: Thomas Young
Thomas Young (1773 – 1829) was another polymath. He was educated at the
universities of Edinburgh, Göttingen, and Cambridge; receiving an MD from
Göttingen in 1796; and opened a practice in 1799 in London. In 1798, while still a
medical student, he worked out how we focus on nearby objects by reshaping the
lens in the eye; and in 1801, found that astigmatism also depended on lens shape.
Portrait of Dr. Thomas Young (1773 – 1829)

Also in that year, he performed the interference experiment (see 4 Interference)
that transformed physics. In 1817, he anticipated Maxwell (see 7 Electromagnetic
Theory) by suggesting that light is a transverse, rather than longitudinal wave like
sound1
. The phenomenon of polarization thus began to make sense; but, as the
physics of this is rather deep, we will not pause here to explain it. Prior to Young, it
was thought that there were separate receptors in the eye for each color; but he
showed that 3 receptors – red, green, and blue – are sufficient. Finally, he made
considerable progress in deciphering the Rosetta Stone, a task completed around
1821 by Jean-François Champollion.
- The Roots
4: Thomas Young
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1
In a sound wave, successive zones of compression and rarefaction advance in the
direction of motion of the wave. In a transverse wave, something changes from side to
side, perpendicular to the direction of the wave. The changes are confined to a “plane
of polarization”.

In studying the motions of the planets (see 2 Laws of Physics), physicists found it
convenient to define the potential energy of a body as the product of the negative
of G (the universal gravitational constant) and the body’s mass, divided by the
distance to another body pulling on it. They also defined the kinetic energy as half
the mass of the body times the square of its speed. It was found that the sum of all
the potential and kinetic energies of all the bodies is fixed. This result is easily
shown to follow directly from Newton’s combined laws of motion and gravity.
- The Roots
5: Conservation of Energy
The gravitational constant G is a key element
in Newton's law of universal gravitation.

5: Conservation of Energy
If one hangs a mass at the end of a spring, and disturbs it, another potential energy
can be defined as proportional to the changing stretch of the spring. Here too, the
sum of the kinetic energy of the mass, and the potential energy of the spring was
found to be constant, independent of the motion of the mass. If the mass is hung
vertically from the spring in the earth’s gravity field, then the sum of the kinetic
energy and the 2 potential energies is now fixed, in spite of any 3 dimensional
motions. This conservation of energy holds whatever the nature of the spring.
Gradually, this invariant total energy was viewed as having a life of its own. It will
be expanded in the next box, and later will play a central role in quantum
mechanics and relativity.
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6: Thermodynamics and Statistical Mechanics
By the late 18th century, heat was recognized as a form of energy, in that it could
cause matter to change temperature, or change state (melting or boiling, or the
reverse). Previously, heat was regarded as a material substance, “caloric”, which
could be added to something to raise its temperature. Caloric was abandoned in
favor of the idea that heat was some form of vibration. This led to the notion that
mechanical energy and heat are related. Indeed, careful experiments showed that
when mechanical energy was converted to heat through friction, or the reverse in a
steam engine, the total is conserved. The concept of the conservation of energy
had been greatly expanded, and became known as “the 1st law of
thermodynamics”.
Joule’s Water-Churning Apparatus
Engraving of Joule's apparatus for
measuring the mechanical
equivalent of heat.

One statement of the First Law of thermodynamics is that there is no process by
which heat may be moved from one reservoir to another at a higher temperature,
without adding energy. This was first clearly stated by Nicolas Léonard Sadi Carnot
(1796 – 1832), who established the maximum efficiency of an engine that converts
heat to mechanical energy, and is sometimes called the Father of Thermodynamics.
Many others contributed to its development (see 6 The Who of Thermodynamics).
In general, a heat engine is a device that transfers heat from a high temperature
reservoir to a lower one, while diverting some of the energy as mechanical,
electrical, chemical, or other usable form.
- The Roots
Heat engine diagram - where heat flows from a high
temperature TH furnace through the fluid of the
"working body" (working substance) and into the cold
sink TC, thus forcing the working substance to do
mechanical work W on the surroundings, via cycles of
contractions and expansions.

Central to the 2nd law of thermodynamics is a mysterious concept called “entropy”.
First, absolute temperature. In 19th century physics, this is where all atomic and
molecular motion would cease. Absolute temperature is measured in kelvins (K). 0
K corresponds to -273 C, or -459 F. So water freezes at 273 K. Now to entropy. In
any heat transfer process, in or out, a tiny change of entropy is given by the
corresponding tiny transfer of heat, divided by the absolute temperature of the
reservoir. A different statement of the 2nd law is that, if you count both the source
of heat and the sink, the sum of the entropy changes is always positive; i.e., total
entropy never decreases.

Viewed somewhat differently, if you add 2 fluids at different temperatures, you get
a mixture at some intermediate temperature; and the total entropy increases. The
2nd law says there is no way to unmix them, as this would decrease the entropy. In
the static universe of the 19th century, this meant that everything would head
toward a single uniform temperature, known then as “the heat death of the
universe”. Well, physicists have argued over this for 150 years, with the waters
muddied by relativity and quantum mechanics; so we aren’t going to settle it here.
A waggish expression of these laws is: 1. You can’t get something for nothing; and
2. You can’t even break even.

Statistical mechanics had its beginning in the researches of Robert Boyle (1627 –
1691). Boyle’s law states that, at constant temperature, the pressure and volume
of a gas are inversely related. In Newtonian mechanics, if you can specify the
position and velocity of each of a set of spherical particles, you can, in principle,
determine their future motions till the end of time. Since this is plainly impractical,
the idea of treating a rich ensemble statistically gained credence, albeit reluctantly.
Portrait of Robert Boyle, c. 1689

The 1st real success along this line was by James Clerk Maxwell (1831 – 1879) (see 7
Maxwell). He was able to derive the distribution of velocities of a large ensemble of
spherical gas molecules, colliding with one another in 3 dimensions, and how it
depended on temperature (see figure on the next slide; there’s much to be gleaned
from it; but, if this is your first look, don’t spend too much time on it). This
development became the basis for understanding such properties as diffusion and
heat transport in gasses. Significant improvements were introduced by Ludwig
Boltzmann (1844 - 1906); so we now call this development the Maxwell –
Boltzmann statistics. Several others contributed to the development of statistical
mechanics, including Josiah Willard Gibbs (1839 – 1903) (see 6 The Who of
Statistical Mechanics). Besides laying the groundwork for a much better
understanding of the behavior of gasses, statistical mechanics caused a revolution
in thought – physicists now started to look at the universe in probabilistic terms.
- The Roots
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What’s plotted is the probability distribution
function of molecular speeds, shown for
hydrogen gas at 3 different Kelvin temperatures.
The peak of each curve is the most likely speed.
As each molecule has some speed, the total
probability (area) under each curve is 1.
Maxwell’s Molecular Speed Distribution in Gasses
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Several people played star roles in the development of thermodynamics in the 19th
century. Of these, arguably the first was Nicolas Léonard Sadi Carnot (1796 - 1832).
Born in France, he entered the École Polytechnique in 1812, and graduated in 1814.
After years in the French army, he turned to physics and chemistry, and became
interested in steam engines. Although others worried mostly about the mechanical
details, Carnot was able to show that the maximum efficiency of a heat engine (the
fraction of heat passing from hot to cold, convertible to useful energy) depends
only on the temperatures of the hot and cold reservoirs. No heat engine, of
whatever design, can exceed the Carnot efficiency. Carnot published this analysis in
1824; but it got little attention till 1834 when Émile Clapeyron, a railroad engineer
extended his result. In 1850, Rudolf Julius Emanuel Clausius (1822 – 1888), a
German physicist generalized this idea further to what is now known as the 2nd Law
of Thermodynamics (see 6 Thermodynamics & Statistical Mechanics). Many
others, including Wilhelm Wien (1864 – 1928), and Hermann von Helmholtz,
(1821 – 1894) contributed to thermodynamics.
- The Roots
6: The Who of Thermodynamics
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The loudest guns in statistical mechanics were James Clerk Maxwell (1831 – 1879)
(see 7 Maxwell), Ludwig Eduard Boltzmann (1844 – 1906), and Josiah Willard Gibbs
(1839 – 1903). Boltzmann received a doctorate from the University of Vienna in
1866, and later held professorships in Vienna, Graz, Munich, and Leipzig. Probably
his main contribution to Physics was to show that the 2nd law of thermodynamics
could be derived from statistical mechanics. He also showed that gasses tended
toward thermodynamic equilibrium because it is by far the most probable state.
This work was widely misunderstood and criticized, presumably leading to his later
depression and suicide. Thereafter, considerable experimental work led to his
vindication.
- The Roots
6: The Who of Statistical Mechanics
Ludwig Eduard Boltzmann (1844-1906)

6: The Who of Statistical Mechanics
Statistical mechanics also profited greatly from the studies of Gibbs, an American
chemist and theoretical physicist. He was educated at Yale and in Europe; and was
appointed Professor of Mathematical Physics at Yale in 1871, a position he held
until his death. He applied the newly emerging statistical mechanics to physical and
chemical processes, and gave the whole field a more rigorous mathematical
foundation. Gibbs was awarded the Copley medal of the Royal Society of London, a
sign that science in Europe was well ahead of the U. S. at the time.
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By the beginning of the 19th century, mechanics had come a long way; but electric
and magnetic phenomena were still a disconnected jumble of observations. Shuffle
your feet on a carpet, and you might draw sparks, and attract bits of paper.
Lodestones attracted bits of iron, and coils of wire carrying an electric current did
the same. Along came Michael Faraday (1791 - 1867), (see 7 Faraday). He
introduced the ideas of lines of force, and electric and magnetic fields. He showed
that 2 current carrying wires exerted forces on each other; and he demonstrated
that a current change in one wire would generate a current change in another.
These findings are the basis of all of today’s transformers, motors, and generators.
- The Roots
7: Electromagnetic Theory

What Faraday lacked in formal mathematical training, James Clerk Maxwell (1831 -
1879), (see 7 Maxwell) possessed in abundance. Gathering what was known of
electric and magnetic phenomena (mostly from Faraday’s work), he produced a set
of equations covering the lot. Realizing that the set appeared incomplete, he made
an inspired guess. Since magnetic field changes induced electric effects, he
supposed that electric field changes would induce magnetic fields. In mathematical
clothes, the complete set is now known as The Maxwell Equations, familiar to every
physicist, and electrical and electronic engineer.
- The Roots

In free space, and lacking any charges or currents, these equations reduce to
separate electric and magnetic disturbances, moving in lockstep. Each field creates
the other as it goes. Technically, such relations are known as “wave equations”,
employed at the time to describe water and sound waves. You will encounter them
again as we get into quantum mechanics, but in entirely different clothes.
Could all this be real? In both wave equations, the speed of propagation is readily
calculated from then known properties of static electric and magnetic fields. This
speed came out to be about 300,000 kilometers per second – the speed of light!
Light is an electromagnetic wave! When all this appeared in various publications in
the 1860’s and 1870’s, the door was opened to today’s technology.

From planetary and solar magnetic fields to cell phones, radio and TV, and
microwave ovens, the Maxwell Equations are all around you.
Moreover, the wave theory of light, so beautifully demonstrated by Thomas
Young’s experiments (see 4 Interference), had been given new meaning. The
unification of electric and magnetic phenomena was the second great coalescence
in physics, after the widening of the conservation of energy. Read on; there will be
others.
There was one loose end – waves of what? When the luminiferous aether was
demolished in the Michelson – Morley experiment of 1887 (see 20 Special
Relativity), it was recognized that none of the supposed properties of the ether
appeared in the Maxwell Equations; so the equations were saved, if not the sanity
of physicists.
- The Roots
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7: Michael Faraday
Son of a blacksmith, Michael Faraday (1791 – 1867) was born south of London.
With little schooling, he was apprenticed to a bookbinder, which gave him the
opportunity to read books brought in for rebinding. An article on electricity in the
third edition of the Encyclopædia Britannica particularly fascinated him; and led him
to simple experiments with electricity and electrochemistry. When offered a ticket
to a lecture by Sir Humphrey Davy (1778 – 1829), a chemist with a considerable
reputation, Faraday went, took extensive notes, and was greatly impressed.
Faraday bound his various notes and sent them off to Davy, along with a request for
a job. When a position opened eventually, Davy hired him as a Laboratory Assistant
in 1812. Later, when asked what was his greatest discovery, Davy promptly replied,
“Michael Faraday”. This second apprenticeship gave Faraday a solid grounding in
chemistry, but ended in 1820.
Portrait of Michael Faraday, c. 1820

Davy had discovered chlorine by dissociating hydrochloric acid, thus showing that
acids didn’t necessarily contain oxygen. Faraday went on to produce compounds of
carbon and chlorine (C2Cl4 and C2Cl6), and was the first to find and describe benzene
(C6H6). We’ll return to benzene when we touch on its peculiar quantum behavior in
22 Modern Chemistry & Molecular Biology.
Following his stint with Davy, Faraday was hired by the Royal Society in London,
lived there, married Sarah Barnard, carried out important research on steel alloys
and high refractivity optical glass, and showed that the latter slightly expels
magnetic fields (diamagnetism). He then returned to electrical phenomena. In
1820, Hans Christian Ørsted (1777 – 1851) found that a magnetic compass always
pointed widdershins (counterclockwise) around a current carrying wire. On hearing
of this, André-Marie Ampère (1775 – 1836) quickly developed a mathematical
theory, which Faraday turned into the first electric motor.
- The Roots
7: Michael Faraday

The principle involved is called electromagnetic induction, where a varying
magnetic field causes a varying current in a conductor, or vice versa. It is the basis
for all electric motors, generators, and transformers; all of which, Faraday was the
first to construct. In the latter, Faraday wound 2 coils of wire on the same iron ring;
a varying current in one coil caused a varying current in the other. The current and
voltage in the 2nd coil depends on the ratio of the number of turns in the 2 coils.
In 1831, Faraday worked with Charles (later Sir Charles) Wheatstone (1802 – 1875)
to investigate sound waves in plates. If one excites the plate with a violin bow to
create transverse waves in the plate, and sprinkles powder on the plate, the
particles will cluster in groups known as Chladni figures. Moreover, a nearby plate
could be excited with similar figures, by sound transmission through the air.
These effects gave rise to Faraday’s induction experiments mentioned above.
7: Michael Faraday

In 1832, he turned to electrochemistry, and found the laws that apply; first, that the
amount of material removed from one plate in a conductive solution, and deposited
on another, depended directly on the current passed through the solution. Second,
if the solution is dissociated, the amount deposited at each plate is in the ratio of
their chemical equivalents (number of atoms divided by valence).
In later years, he carefully experimented on the magnetic properties of various
materials. He found that some slightly enhanced the ambient field, which he
labeled “paramagnetic”, and some that slightly expelled the ambient field, which he
called “diamagnetic”, terms we use to this day. He also developed the idea that
electric and magnetic fields permeated space, ideas that were to inspire James
Clerk Maxwell (1831 – 1879) to dress them in mathematical clothes that were to
revolutionize physics (see 7 Maxwell).
- The Roots
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7: Michael Faraday
Michael Faraday, c. 1861

7: James Clerk Maxwell
James Clerk (pronounced Clark, as in Kent) Maxwell (1831 – 1879), was born in
Edinburgh, Scotland. At age 16 he entered the University of Edinburgh, but
transferred to Cambridge in 1850. Because of his father’s poor health, he returned
to Scotland, and was appointed professor of natural philosophy at Marischal
College, Aberdeen. In 1858, he married Katherine Mary Dewar.
Alas, in 1860, Marischal merged with King’s College, and became the University of
Aberdeen. Maxwell was rejected for tenure, and was required to leave, an event
the University has never lived down. He left behind a laboratory of great quality for
its day; but, unlike the DMNS, it’s short of explanations, and closed to the public.
After this fiasco, Maxwell was appointed professor of natural philosophy and
astronomy at King's College, London. Following the death of his father in 1865,
he returned to Scotland, where he buried himself in research. In 1871, he
moved to Cambridge, as the first professor of experimental physics. He set
up the Cavendish Laboratory there in 1874. He continued in this position until
1879, when illness forced him to resign.

In 1849, his earliest scientific work showed that Thomas Young’s hypothesis that 3
color receptors in the eye are sufficient, is correct (see 4 Young). He did this by
spinning disks with red, green, and blue pie segments, showing that he could
produce everything the eye can see with varying segments of each. Also, in 1861,
he produced what was essentially the 1st color photo, by exposing the same scene
through red, green, and blue filters, and projecting them simultaneously on the
same screen.
During the period 1855 – 9, Maxwell studied the dynamics of Saturn’s rings. He
found that both solid and liquid rings would be gravitationally unstable. He
concluded that they must be icy or rocky rubble, as there were no other viable
options. The Pioneer and Voyager fly-bys of Saturn in the 1970s and 1980s proved
him right.
- The Roots
The first permanent color photograph, taken by
James Clerk Maxwell in 1861. Subject is a Tartan
ribbon.

Around 1860, Maxwell turned to the kinetic theory of gasses. In 1857, Rudolf
Clausius (1822 – 1888) had shown that the molecules of a gas are in motion,
continually bouncing off each other, and had determined the mean free path – the
average distance between collisions. Maxwell looked deeper, and was able to work
out the probability distribution function of molecular speeds (see figure in 6
Thermodynamics & Statistical Mechanics). This function is the probability dP that
the speed v will lie in a very small range dv, as a function of v. The function
depends only on the molecular properties, and on the absolute temperature of the
gas. This was the beginning of statistical mechanics. 8 years later, Ludwig
Boltzmann (1844 – 1906) (see 6 The Who of Statistical Mechanics) extended
Maxwell’s result to cover the conduction of heat in gasses. Today, these works
together are called the Maxwell–Boltzmann statistics – they caused a revolution in
physicist’s view of the universe.
- The Roots

Maxwell is best known today as the father of electromagnetic field theory (see 7
Electromagnetic Theory). That electric and magnetic fields are related was known
to Michael Faraday (1791 – 1867) (see 7 Faraday), who had shown that a changing
magnetic field could induce an electric field in a conductor. Over several years,
starting in 1855, Maxwell found concise mathematical descriptions of Faraday’s
results; but felt that something was missing. He decided that symmetry demanded
that a changing electric field would create magnetic field changes. On stating this
mathematically, his complete set of relations are known today as “The Maxwell
Equations”. Wave solutions of these equations turned out to describe light, radio
waves, X-rays; indeed, the whole electromagnetic spectrum – if you are now in
Space Odyssey, perhaps a third of the exhibits pertain to aspects of electromagnetic
phenomena.
- The Roots

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Maxwell is generally considered the greatest theoretical physicist of the 19th
century. In 1931, on the 100th anniversary of Maxwell's birth, Einstein described
the change in the conception of reality in physics that resulted from Maxwell's work
as “the most profound and the most fruitful that physics has experienced since the
time of Newton”.
James Clerk Maxwell in his 40’s

Till now, our story has concerned the development of physical ideas through the
middle of the 19th century. That something was seriously wrong with this structure
first emerged with the research of Gustav Robert Kirchhoff (1824 – 1887) (see 8
Kirchhoff). He had chosen to investigate the properties of “blackbodies”. In physics,
a black body is defined as something that perfectly absorbs all incident radiation,
and perfectly reemits it. No such stuff exists; but a close approximation is achieved
by holding a hollow box at some uniform temperature. Radiation from the inner
wall impinges on other parts of the wall, is absorbed and reradiated, until some sort
of equilibrium is reached. If a tiny hole is drilled in the box, some radiation will
escape; and its spectrum may be measured. A spectrum is the distribution of
energy, as a function of wavelength. It typically rises to a peak as wavelength
increases, and then falls off.
- The Trunk
8: Blackbody Studies

Kirchhoff showed, experimentally, that the peak, and the shape of the distribution
depends only on the temperature, and not on the box properties, such as size,
shape, or the materials it was made of. Calling attention to this, he suggested that
there must be something very fundamental about blackbody radiation. Little did he
know that he had started a revolution in physics, and in our view of the universe.
Many others followed his lead, confirming his results, and leading to the
unsuccessful theoretical investigations in 10 Ultraviolet Catastrophe.
- The Trunk
8: Blackbody Studies
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Gustav Robert Kirchhoff (1824 – 1887) was born in Königsberg, Prussia (now
Kaliningrad, Russia). In 1845 he first announced Kirchhoff's laws, for the calculation
of the currents and voltages of electrical networks. Extending the theory of the
German physicist Georg Simon Ohm (1789 – 1854), he generalized the equations
describing current flow to the case of networks in three dimensions. In further
studies he demonstrated that current flows through a conductor at nearly the
speed of light.
In 1847 Kirchhoff became Privatdozent (unsalaried lecturer) at the University of
Berlin and three years later accepted the post of extraordinary professor of physics
at the University of Breslau. In 1854 he was appointed professor of physics at the
University of Heidelberg, where he joined forces with Robert Wilhelm Bunsen (see 9
Bunsen) and founded spectrum analysis (see 9 Spectroscopy). Applying this
new research tool, they discovered two new elements, cesium (1860) and
rubidium (1861).
- The Trunk
8: Kirchhoff

The Tree of Quantum Mechanics - The Trunk
Kirchhoff went further to apply spectrum analysis to study the composition of the
sun. He found that when light passes through a gas, the gas absorbs those
wavelengths that it would emit if heated. He used this principle to explain the
numerous dark lines (Fraunhofer lines) in the Sun's spectrum. That discovery
marked the beginning of a new era in astronomy. In 1875 Kirchhoff was appointed
to the chair of mathematical physics at the University of Berlin.
8: Kirchhoff
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Spectroscope of Kirchhoff and Bunsen

The science of spectroscopy may be said to have begun with Newton (see 2 Particle
Theory of Light). With a prism, he spread out sunlight into a rainbow, and showed
that it contained all the colors we can see. Newton’s instrument was improved by
William Hyde Wollaston (1766 – 1828), who in 1802 glimpsed dark lines
superimposed on the rainbow. However, Joseph von Fraunhofer (1787 – 1826)
really made it take off. In 1814 he added a telescope following the prism, and
began to map the lines in the solar spectrum. In 1819, he managed to rule many
parallel fine lines on a plate of glass, to produce what is now called a “diffraction
grating”. Much like the Young 2 slit experiment (see 4 Interference), the grating
causes each wavelength to concentrate at a different spot on a final screen, where
the contributions from each grating line add up in phase. The grating took over in
spectroscopy when Henry Augustus Rowland (1848 – 1901) in 1882 developed
a ruling engine that could produce large, precise, and very densely scribed
gratings. You can view various atomic spectra through diffraction gratings
at our Spectroscopy Exhibit.
- The Trunk
9: Spectroscopy

An essential contribution was made by Gustav Robert Kirchhoff (1824 – 1887) (see 8
Kirchhoff) and Robert Wilhelm Bunsen (1811 – 1899) (see 9 Bunsen). Around 1859,
they showed that, if an element is vaporized, and heated to incandescence, its
spectrum contained a number of bright lines, unique to each element. Moreover, if
a transparent tube, containing a cool gaseous element is placed before a white light
source, dark lines appeared at the same wavelengths as in that element’s emission
spectrum. This showed that the lines seen by Wollaston and von Fraunhofer in
sunlight are caused by absorption and scattering in the atmospheres of the sun and
earth.
- The Trunk
9: Spectroscopy

Kirchhoff and Bunsen developed techniques for analyzing spectra from unknown
sources, and sorting out the lines to determine the source elements, and their
relative proportions. Thus, they were the first to determine the composition of the
sun. Later, improvements in instruments, and the use of large telescopes to gather
sufficient light, allowed spectroscopy to be applied to the stars (see 24
Astrophysics). Today, we have millions of stellar spectra, showing a huge range of
compositions. Many of these were taken with instruments in space, to avoid
absorption and scattering in the earth’s atmosphere, which blocks much of the
spectrum. Most of what we know of stellar structure and dynamics comes from
spectroscopy.
- The Trunk
9: Spectroscopy
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Robert Wilhelm Bunsen (1811 – 1899) received a Ph.D. in chemistry at the
University of Göttingen in 1830, and taught at the universities of Marburg and
Breslau, and at Heidelburg from 1852 until his death. During his career, he was
responsible for many developments in chemical instrumentation; though, oddly,
probably not for the burner that bears his name.
Most important for science was his collaboration with Gustav Kirchhoff (see 8
Kirchhoff). They found that heating a gas to incandescence caused it to emit a set
of wavelengths, or colors, unique to each gas. Moreover, when a cool gas was
placed in front of a white light source, it absorbed or scattered the same set of
wavelengths. This was the true beginning of the science of spectroscopy (see 9
Spectroscopy). By these means, they discovered the elements cesium and
rubidium, both of which were to play important roles in the development of atomic
clocks.
- The Trunk
9: Bunsen
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Theoreticians attempted to derive the blackbody spectrum from the Maxwell
Equations (see 7 Electromagnetic Theory) and thermodynamics (see 6
Thermodynamics & Statistical Mechanics). In 1893, Wilhelm Wien (1864 – 1928)
came up with a formula, based on thermodynamic arguments, that described the
peak and short wavelengths, but failed at long wavelengths. A different approach
from the Maxwell equations by Lord Rayleigh (John William Strutt 1842 – 1919) (see
10 Rayleigh), and later by James Jeans (1877 – 1946), failed completely at short
wavelengths. The Rayleigh-Jeans formula had the intensity going off to infinity at
the violet end, suggesting that the blackbody was emitting infinite energy – plainly
false. Something was clearly awry with 19th century physics and maybe the
Maxwell Equations; the problem became known as “The Ultraviolet Catastrophe”.
Of course, what was in trouble was physics. The fix would require another
revolution – stay tuned.
- The Trunk
10: The Ultraviolet Catastrophe
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John William Strutt, 3rd Baron Rayleigh (1842 – 1919) was born in Langford Grove,
Maldon, Essex, England. In 1861 he entered Trinity College, Cambridge, from which
he graduated with a B.A. in 1865. In his first paper, published in 1869, he gave a
lucid exposition of some aspects of the electromagnetic theory of James Clerk
Maxwell (see 7 Maxwell) in terms of analogies that the average person would
understand.
An attack of rheumatic fever in 1871 threatened his life for a time. On a
recuperative trip up the Nile, he began work on his great book, “The Theory of
Sound”, in which he examined questions of vibrations and the resonance of elastic
solids and gases. The first volume appeared in 1877, followed by a second in 1878,
concentrating on acoustical propagation in material media. After some revision
during his lifetime and successive reprintings after his death, the work has
remained the foremost monument of acoustical literature.
- The Trunk
10: Rayleigh

Perhaps his most significant early work was his theory explaining the blue color of
the sky as the result of scattering of sunlight by small particles in the atmosphere.
The Rayleigh scattering law, which evolved from this theory, has since become
classic in the study of all kinds of wave propagation.
In the period 1879–84, he served as the second Cavendish professor of
experimental physics at Cambridge, after Maxwell. Later he became secretary of
the Royal Society, a post he held for 11 years. Rayleigh's greatest contribution to
science, with William Ramsey (1852 – 1916), is generally considered his discovery
and isolation of argon in 1895, which makes up about 1% of the atmosphere.
10: Rayleigh
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Imagine a glass tube as in the figure, containing 2 electrodes, and a very low
pressure gas. If a potential is applied between the plates, a current will flow, and a
green glow will appear on most of the glass inner surface. This was first done
around 1854 by Johann Heinrich Wilhelm Geissler (1814 - 1879), who had devised a
pump capable of reaching a low enough pressure. The tube was improved by Sir
William Crookes (1832 - 1919). He established that the glow came from something
emitted from the negative electrode or cathode. He did this by introducing various
kinds of obstacles in the tube, which showed that the “cathode rays”, whatever
they were, traveled in straight lines. Since the rays could be deflected by a magnet,
it was evident that they carried a charge, and from the direction of bending, that
the charge is negative. With his modifications, the device became known as a
“Crookes tube”.
11: Discovery of the Electron

In 1897, Sir Joseph John Thomson (1856-1940) (see 11 Thomson) added another
pair of electrodes, orthogonal to those in the figure, and again deflected the
cathode rays. Between these observations, he was able to show that the rays are
particles, around 2000 times less massive than hydrogen atoms, and are now called
“electrons”. That this was a distinct particle was demonstrated because cathode
rays from many different cathode materials are identical.
Since electrons appeared to be constituents of matter, Thomson advanced the
“plum pudding” model – they were embedded in a much heavier ball of positively
charged stuff, a model that was to survive only 16 more years. Thomson received
the Nobel Prize for physics for this work in 1906.
- The Trunk

From the deflection of cathode rays in electric or magnetic fields, Thomson and
others could only determine the ratio of charge to mass of the electrons. However,
around 1910, Robert Andrews Millikan (1868 – 1953) found a truly clever way to
accurately measure the charge alone, thus determining the mass as well.
A fine spray of oil drops is introduced into an upper chamber of the apparatus. The
spray is stopped when one drop falls through a small hole into a lower chamber.
There, a bright light both illuminates the drop, and charges it through the
photoelectric effect in the air surrounding the drop (there’s more on this in 13
Photoelectric Effect). The top and bottom of this lower chamber are plates to
which voltages may be applied. A microscope looks into the chamber horizontally.
The drop is alternately allowed to drop freely, and is pulled upward by a positive
voltage on the upper plate; and its terminal velocity in air is measured.
The radius of the drop, and thus its mass, is determined from the free fall
speed; after which its charge is found from the upward velocity.
- The Trunk

The number of electrons clinging to the drop is initially unknown; but the measured
charge is always close to a multiple of a charge e, the charge of the electron.
Millikan received the Nobel Prize in Physics in 1923.
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Sir Joseph John Thomson (1856 - 1940), was born near Manchester, England, and
educated at Owens College (now part of Victoria University of Manchester) and
Trinity College, University of Cambridge. At Cambridge he taught mathematics and
physics, served as Cavendish Professor of Experimental Physics, and was Master of
Trinity College (1918 - 40). He was also president of the Royal Society (1915 - 20)
and Professor of Natural Philosophy at the Royal Institute of Great Britain (1905 -
18).
For experiments on cathode ray tubes around 1897, he is regarded as the
discoverer of electrons, as the carrier of electrical charge. Thomson was awarded
the 1906 Nobel Prize in physics for his work on the conduction of electricity through
gases, and was knighted in 1908.
11: Thomson
Return
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It may be argued that quantum mechanics originated with the researches of Gustav
Kirchhoff (see 8 Kirchhoff) into the properties of black bodies (see 8 Blackbody
Studies). But it’s quite unlikely that Kirchhoff, a giant of 19th century physics, had
even an inkling of what was to come. To recapitulate, a formula expressing the
black body spectral distribution, and derived from thermodynamic arguments by
Wilhelm Wien (1864 – 1928), failed at long wavelengths; and the Rayleigh – Jeans
formula, based on Maxwell’s Equations, went off to infinity at short wavelengths,
clearly impossible. Many wrestled with this “ultraviolet catastrophe”, without
success (see 10 Ultraviolet Catastrophe).
- The Trunk
12: Discrete Energy Levels

Max Planck (1858 – 1947) (see 12 Planck) carefully considered both arguments; and
noted that the Wien formula fit the known spectral data very well at short
wavelengths; while the Rayleigh – Jeans formula did very well at long wavelengths.
After several years, Planck came up with a formula that fit all the data over the
whole range of measured temperatures. The problem was that it had no physical
justification.
In classical mechanics, an oscillator e.g., a pendulum, a mass hanging from a spring,
etc., may have only a single or a set of natural frequencies; but any amplitude and
thus any energy. However, in 1900, Planck suggested that, at any given frequency ν
any such energy must be restricted to a value hν, or any of its harmonics, for some
then unknown constant h – today, Planck’s constant. That we had never
noticed this was because h is a really teensy number; so the allowable
energies are very closely spaced.
- The Trunk

Applying this notion to the atoms making up the inside of a black box, he argued
that radiation emission or absorption could only change the oscillator from one
allowed energy to another. Thus, if the atoms of the inside of the box are capable
of resonating at some frequency ν, then the radiation flying around inside the box
must be composed of chunks with energies which are multiples of hν. He called
each such chunk a “quantum”, a name that has stuck.
Suppose n and m are positive integers, n greater than m. Then, if an atom has an
energy nhν, it can emit a quantum of energy (or photon) (n – m)hν and drop to the
lower energy mhν. This photon can then be absorbed by a different atom, and
raise its energy by the same amount. Note that Planck’s hypothesis wouldn’t have
worked if the allowed energy levels weren’t evenly spaced. Also note that
emission of a photon doesn’t change the resonant frequency ν.

On applying this restriction to the Maxwell Equations, Planck found that he could
derive his previously found formula for the black body spectrum that precisely
matched the measurements at any given temperature – provided he used the right
value of h. So h was determined in this way; and it has a truly teensy value: 6.626 ×
10-34
joule seconds1
. This result also showed why the black body spectrum doesn’t
depend on the physical and geometric properties of the box. The quantum
revolution had begun; but no one knew where it was headed. Around 1925, one of
the first results of the emerging wave mechanics was this quantization of oscillator
energies (see 17 Matrix & Wave Mechanics). Today, we know that these quantized
energy levels correspond to atomic rotation and vibration states; but it took many
years to work it all out.
- The Trunk
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1
A joule is the unit of energy in the International System of units. It is the energy
consumed by a 1 watt light bulb in 1 second.

Max Karl Ernst Ludwig Planck (1858 – 1947), although a gifted musician, chose to
study physics over music and more classical studies at the age of 17. In 1874 he
attended the University of München, transferring to the University of Berlin in
1877. Returning to München, he received his doctoral degree in 1879, with a
dissertation on the 2nd law of thermodynamics; and in the next year, became a
Lecturer there. In 1885, he went on to the University of Kiel, as Associate Professor.
In 1889, he moved back to the University of Berlin, where he was promoted to Full
Professor in 1892.
12: Planck
Max Planck

All through this period, Planck saw the 2nd law of thermodynamics in absolute
classical terms. However, as the idea of the quantization of energy levels began to
take hold (see 12 Discrete Energy Levels) he found it necessary to see it statistically,
following Boltzmann (see 6 Thermodynamics & Statistical Mechanics). This was
quite wrenching for Planck, as for most others. As one biographer put it, Planck was
a reluctant revolutionary. Planck received the Nobel Prize in physics in 1918. That
it took so long (18 years) is a measure of how slow physicists were to accept the
quantum hypothesis.
- The Trunk
12: Planck
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Although Max Planck’s blackbody spectrum formula fit the data better than
anything else, physicists were slow to accept its quantum basis (see 12 Discrete
Energy Levels). They argued that some other ideas for atomic structure might be
found that would produce a similar formula, and be consistent with classical
physics. Clearly, more evidence was needed. It wasn’t long in coming.
The photoelectric effect – that light or ultraviolet radiation falling on a metallic
surface in vacuum causes electrons to be emitted – was likely first noted by
Heinrich Hertz (1857 – 1894) in 1887. On studying this effect in greater detail,
Philipp Eduard Anton Lenard (1862 – 1947), in about 1902, found that the intensity
of the incident radiation affected only the electron emission rate, not the velocity of
the emitted electrons. Moreover, for each metal, there was a maximum
wavelength, beyond which no electrons were emitted. Within the well
established wave theory of light, to say nothing of the Maxwell Equations, this
made no sense. That more intense light should generate more electrons was
fine; but long wavelength radiation should have been just as effective.
- The Trunk
13: The Photoelectric Effect

It remained for Albert Einstein (1879 - 1955) (see 13 Einstein) to, ah, shine light on
this mystery. In 1905, in one of 4 papers he submitted to the journal The Annals of
Physics, he suggested going back to Newton’s corpuscular theory of light (see 2
Particle Theory of Light), as well as Planck’s quantum theory (see 12 Discrete Energy
Levels). In the latter, the energy of the incident quanta (or photons) is hν, where ν
is the frequency of the light, and h is Planck’s constant. (In any wave, the frequency
ν of passing wave peaks is the speed of propagation divided by the wavelength1
.)
Einstein hypothesized that there is a minimum energy P needed to pry an electron
out of any given surface; so the kinetic energy of the emitted electron is hν minus P.
This neatly accounts for the wavelength limit found by Lenard.
- The Trunk
1
This may be easier to see as the speed of propagation is the frequency times the
wavelength.

Einstein’s paper lent much credence to Planck’s quantum theory; but the question
remained – how could light be both waves and particles, or as some wag later put
it, wavicles? Well, hang in here till 19 Statistical Interpretation, and you’ll find a
resolution, of sorts. Einstein received the Nobel Prize for Physics in 1921 for this
work; no one got it for relativity.
- The Trunk
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Albert Einstein (1879 – 1955) was born in Ulm, Germany of middle class Jews. He
has written that 2 things influenced his early years: a magnetic compass at age 5
that started a fascination with invisible forces, and a geometry book at age 12,
which he “devoured”. In 1894, his father’s business failed; and his parents left for
Milan, Italy to find work, leaving young Albert in a Boarding House, to finish his
schooling. Facing military conscription at 16, he bolted, and found his way to Milan.
As a school dropout and draft dodger, his prospects were poor indeed.
13: Einstein
Albert Einstein, 1921

Fortunately, Einstein could apply directly to the Swiss Federal Polytechnic School
(later the Swiss Federal Institute of Technology) in Zürich without the equivalent of
a high school diploma if he passed its stiff entrance examinations. His marks
showed that he excelled in mathematics and physics, but he failed at French,
chemistry, and biology. Because of his exceptional math scores, he was allowed
into the polytechnic on the condition that he first finish his formal schooling. He
went to a special high school in Aarau, Switzerland, and graduated in 1896. He also
renounced his German citizenship at that time. He was stateless until 1901, when
he was granted Swiss citizenship.
13: Einstein

Einstein would recall that his years in Zürich were some of the happiest of his life.
He met many students who would become loyal friends, such as Marcel
Grossmann, a mathematician, and Michele Besso, with whom he enjoyed lengthy
conversations about space and time. He also met his future wife, Mileva Maric, a
fellow physics student from Serbia. He graduated from the Swiss Federal
Polytechnic School in 1900. He and Mileva had a daughter, Lieserl, in 1902, whose
fate is unknown.
This was the nadir of Einstein’s life. His parents objected to marriage with Mileva,
who he couldn’t support without a job; and his professor wouldn’t recommend him
for employment. His father’s business again went under, and became seriously ill;
but, before dying, he gave consent for marriage with Mileva. He did some tutoring
of children; but even this didn’t last long.
13: Einstein

Things turned up when Grossmann’s father recommended him for a job in the Swiss
Federal Patent Office in Bern. With a small, but steady income, he married Mileva
in 1903. Their children, Hans Albert and Eduard, were born in Bern in 1904 and
1910, respectively.
1905 is often called Einstein's “miracle year” - he published 4 papers in the Annalen
der Physik, each of which would alter the course of modern physics:
1. Über einen die Erzeugung und Verwandlung des Lichtes betreffenden
heuristischen Gesichtspunkt (“On a Heuristic Viewpoint Concerning the Production
and Transformation of Light”), which applied quantum ideas to light in order to
explain the photoelectric effect (see 13 Photoelectric Effect). This paper won the
Nobel Prize for 1921.
- The Trunk
13: Einstein

2. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung
von in ruhenden Flüssigkeiten suspendierten Teilchen (“On the Movement of Small
Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory
of Heat”), in which Einstein offered the first experimental proof of the existence of
atoms. By analyzing the motion of tiny particles suspended in still water, called
Brownian motion, he could calculate the size of the jostling atoms and Avogadro's
number, the number of molecules in a mass of the water corresponding to its
molecular weight.
3. Zur Elektrodynamik bewegter Körper (“On the Electrodynamics of Moving
Bodies”), in which Einstein laid out the mathematical theory of special relativity (see
20 Special Relativity).
- The Trunk
13: Einstein

4. Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (“Does the
Inertia of a Body Depend Upon Its Energy Content?”), submitted almost as an
afterthought, which showed that relativity theory led to E = mc2
. This provided the
first mechanism to explain the energy source of the Sun and other stars.
Einstein also submitted a paper in 1905 for his doctorate at the Swiss Federal
Polytechnic School, on the sizes of molecules. There has never been a year like
1905, in the whole history of physics.
At first Einstein's 1905 papers were ignored. This began to change after he received
the attention of just one physicist, perhaps the most influential physicist of his
generation, Max Planck (see 12 Planck).
- The Trunk
13: Einstein

Soon, owing to Planck's comments, and to experiments that gradually confirmed his
theories, Einstein was invited to lecture at international meetings, and he rose
rapidly. He was offered a series of positions at increasingly prestigious institutions,
including the University of Zürich, the University of Prague, the Swiss Federal
Institute of Technology, and finally the University of Berlin, where he served as
director of the Kaiser Wilhelm Institute for Physics from 1913 to 1933.
In 1915, Einstein completed his General Theory of Relativity, extending the Special
Theory to cover accelerated systems and gravitation. Unfortunately, while General
Relativity is the best theory of things in the large, available to physics at this writing,
it’s incompatible with quantum mechanics; so we won’t offer a discussion of it in
this program; but there will be some further mention of it in 28 Theory of
Everything.
- The Trunk
13: Einstein

Einstein's work was interrupted by World War I. A lifelong pacifist, he was only one
of 4 intellectuals in Germany to sign a manifesto opposing Germany's entry into
war. Disgusted, he called nationalism “the measles of mankind”. He would write,
“At such a time as this, one realizes what a sorry species of animal one belongs to”.
There are many good sources of information on the remainder of Einstein’s life;
but, as he made few further contributions to quantum mechanics, we’ll keep it
short. After the bending of starlight as it passes the sun was experimentally
confirmed in 1919, Einstein was universally acclaimed. Sir Arthur Eddington, who
led the expedition to measure this, was later interviewed by a media person, who
said that he’d heard that there were only 3 people in the world who understood
Einstein; to which Eddington, no shrinking violet, promptly replied, “Who’s the
third?”.
13: Einstein

Inevitably, Einstein's fame and the great success of his theories created a backlash.
The rising Nazi movement found a convenient target in relativity, branding it
“Jewish physics” and sponsoring conferences and book burnings to denounce
Einstein and his theories. The Nazis enlisted other physicists, including Nobel
laureates Philipp Lenard and Johannes Stark, to denounce Einstein. One Hundred
Authors Against Einstein was published in 1931. When asked to comment on this
denunciation of relativity by so many scientists, Einstein replied that to defeat
relativity one did not need the word of 100 scientists, just one fact.
In December 1932 Einstein decided to leave Germany - he would never go back. It
became obvious to Einstein that his life was in danger. A Nazi organization
published a magazine with Einstein's picture and the caption “Not Yet Hanged” on
the cover. There was even a price on his head. So great was the threat that
Einstein split with his pacifist friends and said that it was justified to defend
yourself with arms against Nazi aggression. To Einstein, pacifism was not
an absolute concept but one that had to be re-examined depending
on the magnitude of the threat.
13: Einstein

Einstein settled at the newly formed Institute for Advanced Study at Princeton, N.J.,
which soon became a mecca for physicists from around the world. Newspaper
articles declared that the “pope of physics” had left Germany and that Princeton
had become the new Vatican.
To his horror, during the late 1930s, physicists began seriously to consider whether
his equation E = mc2
might make an atomic bomb possible. In 1920 Einstein himself
had considered, but eventually dismissed the possibility. However, he left it open, if
a method could be found to release the power of the atom. Then in 1938–39 Otto
Hahn, Fritz Strassmann, Lise Meitner, and Otto Frisch showed that vast amounts of
energy could be unleashed by the splitting of the uranium atom. The news
electrified the physics community.
13: Einstein

In July 1939 physicist Leo Szilard asked Einstein if he would write a letter to U.S.
President Franklin D. Roosevelt urging him to develop an atomic bomb. Following
several translated drafts, Einstein signed a letter on August 2 that was delivered to
Roosevelt by one of his economic advisers, Alexander Sachs, on October 11.
Roosevelt wrote back on October 19, informing Einstein that he had organized the
Uranium Committee to study the issue.
13: Einstein
Scan of the letter sent to U.S. President
Franklin D. Roosevelt on August 2, 1939

Einstein was granted permanent residency in the United States in 1935 and became
an American citizen in 1940, although he chose to retain his Swiss citizenship.
During the war, Einstein's colleagues were asked to journey to Los Alamos, N.M., to
develop the first atomic bomb for the Manhattan Project. Einstein, the man whose
equation had set the whole effort into motion, was never asked to participate.
Voluminous declassified FBI files, numbering several thousand, reveal the reason:
the U. S. government feared Einstein's lifelong association with peace and socialist
organizations. FBI director J. Edgar Hoover went so far as to recommend that
Einstein be kept out of America by the Alien Exclusion Act, but he was overruled by
the U.S. State Department. Instead, during the war Einstein was asked to help the
U.S. Navy evaluate designs for future weapons systems. Einstein also helped the
war effort by auctioning off priceless personal manuscripts. In particular, a
handwritten copy of his 1905 paper on special relativity was sold for $6.5 million.
It’s now located in the Library of Congress.
13: Einstein
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The early years of the 20th century were to shake physics to its core. Major
palpitations were caused by Ernest Rutherford, 1st Baron Rutherford of Nelson and
Cambridge (1871 - 1937). Radioactivity had been discovered by Henri Becquerel
(1852 – 1908) in 1896. It was found to be composed of 3 kinds of radiation, called
alpha, beta, and gamma. Beta radiation was found to be Thomson’s electrons; and
gamma was eventually identified as extremely short wavelength electromagnetic
radiation.
In 1909, Rutherford placed some radioactive material inside a thin walled
evacuated glass tube, surrounded by another thicker wall, also evacuated. The
inner wall stopped the electrons; but the alphas penetrated to the space between
the walls. An electric discharge in that space caused a weak glow which showed the
spectral lines of helium. Since alphas would bend in a magnetic field,
they were identified as helium atoms, with their 2 electrons removed.
14: Discovery of the Nucleus

In 1913, he set one of his students to measure alpha scattering by a thin platinum
foil. To their great surprise, about one in 8000 was deflected by more than 90
degrees. As he put it, “it was almost as incredible as if you had fired a 15-inch shell
at a piece of tissue-paper and it came back and hit you”. What hit physics was that
the “plum pudding” model of the atom (see 11 Discovery of Electrons) had just
expired. What grew up in its place was a hard positively charged core, now called
the nucleus, containing almost all of the atom’s mass, surrounded by a bunch of
electrons. For his earlier work, he was awarded the Nobel Prize in Chemistry.
- The Trunk
Top: Expected results: alpha particles passing
through the plum pudding model of the atom
undisturbed.
Bottom: Observed results: a small portion of
the particles were deflected, indicating a small,
concentrated positive charge.

The fallout from this discovery was to be profound; for it presented physics with an
enduring mystery – what holds nuclei together? The simplest atom, hydrogen,
consists of one positive proton, with one electron orbiting about it. The next,
helium, has an atomic weight about 4 times larger, and was assumed to consist of 4
protons, plus 2 electrons in the nucleus, and 2 more orbiting about it, thus creating
a neutral atom. More massive nuclei were assumed to be built up around the same
plan.
So, now dressed in more elaborate clothes, what holds nuclei together? All but
hydrogen ought to fly apart. Physicists of the time posited a “strong force” did the
job; but this term only labeled their ignorance. In 26 Nuclear Physics, we’ll take this
up again.
- The Trunk
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Ernest Rutherford, 1st Baron Rutherford of Nelson and Cambridge (1871 - 1937)
was born in Brightwater, New Zealand. In 1887 he won a scholarship to Nelson
College, a secondary school. He won prizes in history and languages as well as
mathematics. Another scholarship allowed him to enroll in Canterbury College,
Christchurch, from which he graduated with a B.A. in 1892 and an M.A. in 1893 with
first-class honors in mathematics and physics. Financing himself by part-time
teaching, he stayed for a fifth year to do research in physics, studying the properties
of iron in high-frequency alternating magnetic fields. He found that he could detect
the electromagnetic waves, newly discovered by Heinrich Hertz (1857 – 1894), even
after they had passed through brick walls. Two substantial scientific papers on this
work won for him an “1851 Exhibition” scholarship, which provided for further
education in England.
14: Rutherford

On his arrival in Cambridge in 1895, Rutherford began to work under J. J. Thomson
(see 11 Thomson), professor of experimental physics at the university's Cavendish
Laboratory. Toward the end of the 19th century many scientists thought that no
new advances in physics remained to be made. Yet within 3 years Rutherford
succeeded in marking out an entirely new branch of physics called radioactivity. He
soon found that thorium or its compounds disintegrated into a gas that in turn
disintegrated into an unknown “active deposit”, likewise radioactive.
- The Trunk
14: Rutherford
Ernest Rutherford

Rutherford and a young chemist, Frederick Soddy (1877 – 1956), then investigated 3
groups of radioactive elements: radium, thorium, and actinium. They concluded in
1902 that radioactivity was a process in which atoms of one element spontaneously
disintegrated into atoms of an entirely different element, which also remained
radioactive. This interpretation was opposed by many chemists who held firmly to
the concept of the indestructibility of matter; the suggestion that some atoms could
tear themselves apart to form entirely different kinds of matter was to them a
remnant of medieval alchemy. Nevertheless, Rutherford's outstanding work won
him recognition by the Royal Society, which elected him a fellow in 1903 and
awarded him the Rumford medal in 1904.
14: Rutherford

In 1898 Rutherford was appointed to the chair of physics at McGill University in
Montreal, Canada. In 1907 he returned to England to accept a chair at the
University of Manchester, where he continued his research on the alpha particle.
With the ingenious apparatus that he and his research assistant, Hans Geiger (1882
– 1945), had invented, they counted the particles as they were emitted one by one
from a known amount of radium; and they also measured the total charge
collected, from which the charge on each particle could be detected.
14: Rutherford
Johannes (Hans) Wilhelm Geiger

Combining this result with the rate of production of helium from radium,
determined by Rutherford and the American chemist Bertram Borden Boltwood
(1870 – 1927), Rutherford was able to deduce Avogadro's number (the constant
number of molecules in the molecular weight in grams of any substance) in the
most direct manner conceivable1
. With his student Thomas D. Royds he proved in
1908 that the alpha particle really is a helium atom, by allowing alpha particles to
escape through the thin glass wall of a containing vessel into an evacuated outer
glass tube and showing that the spectrum of the collected gas was that of helium.
Almost immediately, in 1908, came the Nobel Prize—but for chemistry, for his
investigations concerning the disintegration of elements. As Rutherford put it, “It
was very unexpected, and I am startled by my metamorphosis into a chemist”.
14: Rutherford
1
About this time, Einstein found an entirely different way to measure Avogadros number, see 13
Einstein.

In 1911 Rutherford made his greatest contribution to science with his nuclear
theory of the atom. He had observed in Montreal that fast-moving alpha particles,
on passing through thin plates of mica, produced diffuse images on photographic
plates; whereas a sharp image was produced when there was no obstruction to the
passage of the rays. He considered that the particles must be deflected through
small angles as they passed close to atoms of the mica; but calculation showed that
an electric field of 100,000,000 volts per centimeter was necessary to deflect such
particles traveling at 20,000 kilometers per second, a most astonishing conclusion.
This phenomenon of scattering was found in the counting experiments with Geiger;
Rutherford suggested to Geiger and a student, Ernest Marsden, that it would be of
interest to examine whether any particles were scattered backward—i.e., deflected
through an angle of more than 90 degrees.
14: Rutherford

To their astonishment, a few particles in every 10,000 were indeed so scattered,
emerging from the same side of a gold foil from where they had come. After a
number of calculations, Rutherford came to the conclusion that the intense electric
field required to cause such a large deflection could occur only if all the positive
charge in the atom, and therefore almost all the mass, were concentrated on a very
small central nucleus some 10,000 times smaller in diameter than that of the entire
atom. The positive charge on the nucleus would therefore be balanced by an equal
charge on all the electrons distributed somehow around the nucleus. This theory of
atomic structure is known as the Rutherford atomic model.
14: Rutherford

During World War I he worked on the practical problem of submarine detection by
underwater acoustics. He produced the first artificial disintegration of an element
in 1919, when he found that on collision with an alpha particle an atom of nitrogen
was converted into an atom of oxygen and an atom of hydrogen. The same year he
succeeded Thomson as Cavendish professor. His influence on research students
was enormous. In the second Bakerian Lecture he gave to the Royal Society in 1920,
he speculated upon the existence of the neutron and of isotopes of hydrogen and
helium; 3 of them were eventually discovered in the Cavendish Laboratory.
His service as president of the Royal Society (1925 – 1930), and as Chair of the
Academic Assistance Council, helped almost 1,000 university refugees from
Germany, but increased the claims upon his time. Whenever possible he worked in
the Cavendish Laboratory, where he encouraged students, probed for the facts, and
always sought an explanation in simple terms. In 1931, he was made a Peer.
14: Rutherford
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If physics was in shambles, following the work of Planck and Einstein, the discovery
of the atomic nucleus by Rutherford and his students (see 14 Discovery of the
Nucleus) managed to make matters worse. If the electrons are outside the nucleus,
shouldn’t they be drawn in by the attraction between opposite charges? Yes, but
they could avoid this fate if they are in orbit around the nucleus, much like the
planets around the sun. After all, electrostatic attraction between charges is an
inverse square force, just like gravity; so circular or elliptical orbits would make
sense.
The fly in this ointment is that, in any Keplerian orbit, there is a force on the orbiting
body to keep it from flying off on a straight line. Thus, an orbiting body is being
constantly accelerated. However, the Maxwell Equations say that any
accelerated charge must radiate electromagnetic waves; so the electrons
would lose energy, and spiral into the nucleus. As atoms appear to be stable,
there must be something wrong with this picture.
- The Trunk
15: Bohr Hydrogen Atom

It fell to Niels Henrik David Bohr (1885 - 1962) (see 15 Bohr) to lead the way to a
solution. An essential clue came from spectroscopy (see 9 Spectroscopy) – lines in
the spectrum of every element showed that atoms were only capable of emitting or
absorbing energy at a discrete set of wavelengths. From the new quantum
viewpoint, this suggested that the electrons in an atom could only exist in a finite
set of energy states; and that emission or absorption of a photon implied that an
electron had jumped from one allowed state to another.
The simplest atom is hydrogen, consisting of a single electron, and a nucleus of
equal and opposite charge. From the work of Johann Jakob Balmer (1825 - 1898),
and Johannes Robert Rydberg (1854 – 1919), the lines of the hydrogen spectrum
were known to obey a simple formula involving 2 integers and a constant R,
now called the Rydberg constant. From analysis of the spectrum, the value
of R was known.
- The Trunk

Bohr assembled the above arguments, added a bit of arm waving, derived the
Balmer-Rydberg formula, and demonstrated that R could be calculated from several
known physical constants, including the mass and charge of the electron and
Planck’s constant h. This tour de force was published in a set of papers between
1913 and 1915. With the help of Arnold Johannes Wilhelm Sommerfeld (1868 -
1951), Bohr generalized this scheme somewhat to build up a picture of what more
complex atoms looked like.
As Bohr saw it, hydrogen’s electron had to occupy one of a set of energy states,
signified by a set of integers 1, 2, 3, . . . When emission or absorption occurred, and
the electron jumped between 2 of these states; the corresponding integers are
those appearing in the Balmer-Rydberg formula. A diagram of these energy
states, and transitions between them, is shown below, and may be helpful.

Some confirmation of this picture came from helium, an atom with 2 electrons, and
for which no relation corresponding to the Balmer-Rydberg formula is known.
However, if one of these electrons is somehow removed (ionized in the vernacular),
the spectrum of the resulting atom looks like that of hydrogen, though displaced in
wavelength. Moreover, it obeys the Balmer-Rydberg formula, with a different value
of R, calculable from Bohr’s theory.
The trouble with this “old quantum theory”, as it’s now called, is that it had no firm
theoretical underpinning. Arguments raged everywhere about what all this meant,
with Bohr leading the way from his institute in Kobenhavn; but it would be another
decade before a more comprehensive quantum theory would begin to take shape.
Please don’t jump to 17 Matrix & Wave Mechanics until we throw in some more
confusion.
- The Trunk
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Niels Henrik David Bohr (1885 - 1962) was born in Kobenhavn of Jewish parents. He
attended the university there, where he earned a doctorate in 1911. He then went
to work for J. J. Thomson (see 11 Thomson) at Cambridge, and then Ernest
Rutherford (see 14 Rutherford) in Manchester. It was there that he developed his
theory of the hydrogen atom (see 15 Hydrogen Atom Model). Bohr returned to
Kobenhavn from Manchester in 1912, married Margrethe Nørlund, and continued
to develop his new approach to the physics of the atom. The work was completed
in 1913.
- The Trunk
15: Bohr
Niels Henrik David Bohr

In 1916, after serving as a lecturer in Kobenhavn, and then in Manchester, Bohr was
appointed to a professorship in his native city. The university created for Bohr a
new Institute of Theoretical Physics, which opened its doors in 1921; he served as
director for the rest of his life. Somewhere around 1920 he developed what he
called the “correspondence principle” – essentially that, in reactions involving
radiation, the frequencies calculated from quantum ideas should approach a
classical electrodynamics limit, as the number of atoms involved increases; i.e., that
everyday behavior should evolve directly from quantum behavior at the atomic
level. Bohr's institute soon became an international center for work on atomic
physics and quantum theory.
He also urged physicists to view atomic phenomena from the standpoint of what he
called the “principle of complementarity”, that is, whether you see waves
or particles depends on how you choose to look at them; but you never see
both at once. A complete understanding of what’s going on requires you
to consider both viewpoints.
15: Bohr

Einstein greatly admired Bohr's early work, referring to it as “the highest form of
musicality in the sphere of thought”, but he never accepted Bohr's claim that
quantum mechanics was the “rational generalization of classical physics” demanded
for the understanding of atomic phenomena. Einstein and Bohr discussed the
fundamental questions of physics on a number of occasions, but they never came
to basic agreement. In his account of these discussions, however, Bohr emphasized
how important Einstein's challenging objections had been to the evolution of his
own ideas and what a deep and lasting impression they had made on him.
Bohr's institute continued to be a focal point for theoretical physicists until the
outbreak of World War II. The annual conferences on nuclear physics as well as
formal and informal visits of varied duration brought virtually everyone concerned
with quantum physics to Kobenhavn at one time or another.
15: Bohr

When Denmark was overrun by the Germans in 1940, Bohr did what he could to
maintain the work of his institute and to preserve the integrity of Danish culture
against Nazi influences. In 1943, under threat of immediate arrest because of his
Jewish ancestry, and the anti-Nazi views he made no effort to conceal, Bohr,
together with his wife and some other family members, was transported to Sweden
by fishing boat in the dead of night by the Danish resistance movement. A few days
later the British government sent an unarmed Mosquito bomber to Sweden, and
Bohr was flown to England in a dramatic flight that almost cost him his life. During
the next two years, Bohr and one of his sons, Aage (who later followed his father's
career as a theoretical physicist, director of the institute, and Nobel Prize winner in
physics), took part in the projects for making a nuclear fission bomb. They worked
in England for several months and then moved to Los Alamos, N.M., U.S.,
with a British research team.
15: Bohr

Bohr's concern about the terrifying prospects for humanity posed by such atomic
weapons was evident as early as 1944, when he tried to persuade British Prime
Minister Winston Churchill and U.S. President Franklin D. Roosevelt of the need for
international cooperation in dealing with these problems. Although this appeal did
not succeed, Bohr continued to argue for rational, peaceful policies, advocating an
“open world” in a public letter to the United Nations in 1950. Bohr was convinced
that free exchange of people and ideas was necessary to achieve control of nuclear
weapons. He led in promoting such efforts as the First International Conference on
the Peaceful Uses of Atomic Energy, held in Geneva (1955), and in helping to create
the European Council for Nuclear Research (CERN). Among his many honors, Bohr
received the first U. S. Atoms for Peace Award in 1957.
15: Bohr

In his last years Bohr tried to point out ways in which the idea of complementarity
could throw light on many aspects of human life and thought. He had a major
influence on several generations of physicists, deepening their approach to their
science and to their lives. Bohr himself was always ready to learn, even from his
youngest collaborators. He drew strength from his close personal ties with his
coworkers and with his sons, his wife, and his brother. Profoundly international in
spirit, Bohr was just as profoundly Danish, firmly rooted in his own culture. This
was symbolized by his many public roles, particularly as president of the Royal
Danish Academy from 1939 until his death.
15: Bohr
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Niels Bohr (1885-1962) appears
on the Danish 500 Kroner note

If Bohr had physics in turmoil, matters didn’t appear to improve when, in 1923,
Louis Victor, Prince de Broglie (1892 - 1987), a French graduate student in physics
came up with a really outlandish idea. He said that, in Bohr’s hydrogen atom, the
electron is really a standing wave. In the ground state, there is just one wave
wrapped around the orbit; and each higher energy state has one more wave.
Almost everyone, including Bohr, thought this was nonsense; but Einstein said he
was on to something; et voila!, de Broglie got his Ph. D. In de Broglie’s view, the
wavelength was the circumference of Bohr’s ground state or lowest energy orbit.
The wavelength could also be calculated by dividing h by Bohr’s value of the
electron’s momentum (the product of mass and speed); so faster or more massive
particles have shorter wavelengths.
- Branches
16: Electrons are Waves

Duality was now complete – both light and matter are both waves and particles.
This schizophrenia received experimental confirmation only 4 years later (see 18
Electron Diffraction). It also inspired the first solid attempts to put quantum
mechanics on a firm mathematical footing (see 17 Matrix & Wave Mechanics).
Persevere to 19 Statistical Interpretation, and the wave – particle mess may make
some sense (we fondly hope).
- Branches
16: Electrons are Waves
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De Broglie’s vision of Bohr’s atom

The Tree of Quantum Mechanics - Branches
Louis-Victor-Pierre-Raymond, 7e
duc de Broglie (1892 - 1987) was the second son of
a member of the French nobility. The Broglie family name is taken from a small
town in Normandy. In choosing science as a profession, Louis de Broglie broke with
family tradition, as had his brother Maurice (from whom, after his death, Louis
inherited the title of duc). Maurice, also a physicist, made notable contributions to
the experimental study of the atomic nucleus, and kept a well-equipped laboratory
in the family mansion in Paris. Louis occasionally joined his brother in his work, but
it was the purely conceptual side of physics that attracted him. He described
himself as “having much more the state of mind of a pure theoretician than that of
an experimenter or engineer, loving especially the general and philosophical
view. . . .” He was brought into one of his few contacts with the technical aspects of
physics during World War I, when he saw army service in a radio station in the Eiffel
Tower.
16: De Broglie

De Broglie's interest in what he called the “mysteries” of atomic physics —unsolved
conceptual problems of the science — was aroused when he learned from his
brother about the work of the German physicists Max Planck and Albert Einstein;
but the decision to take up the profession of physicist was long in coming. He
began at 18 to study theoretical physics at the Sorbonne, but he was also earning
his degree in history (1909), thus moving along the family path toward a career in
the diplomatic service. After a period of severe conflict, he declined the research
project in French history that he had been assigned, and chose for his doctoral
thesis a subject in physics.
16: De Broglie
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Louis de Broglie, 1929

In this thesis (1924) de Broglie developed his revolutionary theory of electron
waves, which he had published earlier in scientific journals. The notion that matter
on the atomic scale might have the properties of a wave was rooted in a proposal
Einstein had made 20 years before. Einstein had suggested that light of short
wavelengths might under some conditions be observed to behave as if it were
composed of particles. The dual nature of light, however, was just beginning to
gain scientific acceptance when Broglie extended the idea of such a duality to
matter.
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De Broglie's proposal answered a question that had been raised by calculations of
the motion of electrons within the atom. Experiments had indicated that the
electron must move around a nucleus and that, for reasons then obscure, there are
restrictions on its motion. De Broglie's idea of an electron with the properties of a
wave offered an explanation of the restricted motion. A wave confined within
boundaries imposed by the nuclear charge would be restricted in shape and, thus,
in motion, for any wave shape that did not fit within the atomic boundaries would
interfere with itself and be canceled out. In 1923, when Broglie put forward this
idea, there was no experimental evidence whatsoever that the electron, the
corpuscular properties of which were well established by experiment, might under
some conditions behave as if it were radiant energy. De Broglie's suggestion, his
one major contribution to physics, thus constituted a triumph of intuition.
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The first publications of de Broglie's idea of “matter waves” had drawn little
attention from other physicists, but a copy of his doctoral thesis chanced to reach
Einstein, whose response was enthusiastic. Einstein stressed the importance of de
Broglie's work both explicitly and by building further on it. In this way the Austrian
physicist Erwin Schrödinger (see 17 Heisenberg and 17 Schrödinger) learned of the
hypothetical waves, and on the basis of the idea, he constructed a mathematical
system, wave mechanics (see 17 Matrix & Wave Mechanics), that has become an
essential tool of physics. Not until 1927, however, did Clinton Davisson and Lester
Germer (see 18 Electron Diffraction) in the United States and George Thomson in
Scotland find the first experimental evidence of the electron's wave nature.
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After receiving his doctorate, de Broglie remained at the Sorbonne, becoming in
1928 professor of theoretical physics at the newly founded Henri Poincaré Institute,
where he taught until his retirement in 1962. He also acted, after 1945, as an
adviser to the French Atomic Energy Commissariat.
In addition to winning the Nobel Prize for Physics in 1929, de Broglie received the
Kalinga Prize in 1952, awarded by the United Nations Economic and Social Council,
in recognition of his writings on science for the general public. He was a foreign
member of the British Royal Society, a member of the French Academy of Sciences,
and, like several of his forebears, a member of the Académie Française.
16: De Broglie
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The insights of Planck, Einstein, Bohr, and de Broglie had combined to turn physics
on its head; but 2 experimental results added to the confusion. Around 1921,
Arthur Holly Compton (1892 – 1962) aimed an X-ray beam at a carbon sample,
producing ejected electrons and lower energy X-rays. Since both energy and
momentum were conserved in the process, it was evident that an X-ray photon (a
term invented by Compton) gave some of its energy to the electron, and went off
(weakened) in a different direction. Classically, the X-ray could scatter, but not
change energy; so quantum mechanics had a new playground. For this “Compton
Effect”, he got the Nobel Prize in 1927.
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17: Matrix & Wave Mechanics
A photon of wavelength
λ comes in from the
left, collides with a
target at rest, and a new
photon of wavelength λ
emerges at an angle θ .

About a year later, Otto Stern (1888 – 1969) and Walther Gerlach (1889 – 1979)
passed a beam of silver atoms through a spatially varying magnetic field.
Classically, a magnet experiences both a force and a torque in such a field; so, if the
beam had been composed of itty-bitty magnets, it should have spread out
corresponding to the initial orientations of the magnets. Instead, the silver atoms
neatly divided into 2 well defined beams, even though silver isn’t magnetic.
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Basic elements of the Stern–Gerlach
experiment.

Insight was not long in coming. A pair of Dutch grad students in physics, Samuel
Abraham Goudsmit (1902 – 1978) and George Eugene Uhlenbeck (1900 – 1988),
following a suggestion by P. A. M. Dirac (see 21 Dirac), put forth the idea that an
electron has an intrinsic angular momentum (spin), which can only take on the
values +1/2 and -1/2. They had thus added a new quantum number to the previous
3. Since a spinning charge generates a magnetic field (known since Maxwell), an
electron has a magnetic moment, corresponding to one of its possible spin values.
Later, it was found that protons and neutrons have similar spins, and thus magnetic
moments. Mostly spins pair up and their angular momenta and magnetic moments
cancel out; but both common isotopes of silver have 1 unpaired electron, and thus
each atom has an overall magnetic moment, accounting for its strange behavior in
the Stern - Gerlach experiment.
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In a silver crystal, atoms with opposite spins will pair up; so, in bulk, it has no
magnetic moment, the magnetic properties showing up only in beams of single
atoms. The choice of silver was a lucky one for science; and it got Stern the Nobel
Prize in 1943.
In 1925, inspired by de Broglie’s insight, and the earlier cornerstones laid by Planck,
Einstein, and Bohr; Werner Heisenberg (1901 - 1976) (see 17 Heisenberg),
constructed the first rigorous theory of quantum mechanics. With it, he rederived
Bohr’s results, and widened the theory well beyond hydrogen atoms. Heisenberg
called it “matrix mechanics”. A year later, Erwin Schrödinger (1887 - 1961) (see 17
Schrödinger), unsatisfied with matrix mechanics, even derogatory, returned to de
Broglie’s waves.
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With the advice of David Hilbert (1862 - 1943), the leading mathematician of the
time, Schrödinger recast these ideas in the form of a “partial differential equation”
(don’t worry about it), and again rederived Bohr’s results. He called his scheme
“wave mechanics”. He and Heisenberg sniped at each other for a while; but he
eventually demonstrated that the 2 theories are mathematically equivalent, and
would give the same result to any question you could pose to them. Alas, if you’re
determined to wade around in matrices or partial differential equations, you’ll need
a few years of mathematical training.
Two general results quickly arose from the new theory. One was Wolfgang Pauli’s
(1900 – 1958) (see 17 Pauli) derivation of the “exclusion principle”. This stated that
no 2 electrons in an atom could be in the same state, i.e., they couldn’t share all 4
quantum numbers (see 17 Periodic Table). This arises directly from Schrödinger’s
equation, with the proviso that electrons are all identical particles, and are thus
indistinguishable. It was an immense help in working out the electronic
structure of atoms.
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The term “indistinguishable” needs further comment. In classical mechanics, 2
identical particles might be told apart by adding a red dot to one, and a blue dot to
the other, when you could follow each one around. There would be a separate
equation covering each one’s motion; and if there is a force between them, an
appropriate term appears in each equation. However, in quantum mechanics, only
spin can distinguish 2 electrons, when an exchange of electrons might not be
detectable, even in principle. A joint Schrödinger equation would then describe the
evolution of the system, dissolving into separate equations only if there is no
possible interaction between particles, and remaining unchanged if the 2 particles
are interchanged. This constraint has no parallel in classical physics; and it puts a
tight straitjacket on solutions to the Schrödinger equation.
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The other result was Heisenberg’s uncertainty principle. This states that there is a
fundamental uncertainty in determining the whereabouts of particles (or waves).
For example, if you simultaneously measure the position and velocity of an
electron, then the uncertainty in the position times the uncertainty in momentum2
can’t be less than h/(4π), whatever the accuracy of your measurements. We don’t
notice this in everyday life – if you measure the position and velocity of a baseball,
atomic sized uncertainties are way below what your instruments can deliver.
Uncertainty is intrinsic; but it’s generally because any form of measurement will
disturb what you’re trying to measure. Some other products of pairs of
uncertainties subscribe to this same minimum – understanding which ones and why
is every physics student’s bane.
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2
Recall that the momentum of a particle is its mass times its velocity.

The uncertainty principle isn’t completely new. Consider a very long rope, with you
at one end. If you shake the rope up and down and stop, a wavelike disturbance
will propagate down the rope. If asked what is the wavelength, you might measure
the distance between crests, and take an average. On the other hand, you would
be baffled if asked where this wave is. However, if you had only pulled the rope up
and down once, only one pulse would travel down the rope. Then, the position
question would seem sensible; but the wavelength would not.
Translating this into de Broglie’s terms, the momentum of a moving particle is given
by h divided by the wavelength. So, the uncertainty in momentum is directly
related to that of wavelength, and the analogy is complete. What’s new is the
irreducible minimum uncertainty. In 19 Statistical Interpretation, we’ll try to show
what all this means.
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Shortly after the controversy between Heisenberg and Schrödinger was settled,
physicists decided that Schrödinger’s approach is much easier to apply; and today,
quantum mechanics texts tend to start from Schrödinger’s equation, and explore it
to derive all sorts of physical behavior. However, in only the very simplest cases is it
possible the solve Schrödinger’s equation “by hand”. Generally, a numerical
solution by computer is needed, which can involve a formidable amount of
arithmetic. This is true of every atom more complex than hydrogen. With today’s
computer capabilities, every atom has been solved; and all the electronic energy
levels have been confirmed by spectroscopy (see 9 Spectroscopy), and the periodic
table of elements is now fully understood. Heisenberg had many discussions with
Bohr. As he recounted later, he emerged thinking (more or less) “can the universe
really be like this?”.
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One of the first applications of Schrödinger’s equation was to what physicists call
the one dimensional harmonic oscillator. As an example, consider a mass hanging
from a frictionless spring. Classically, disturb it, and it will bob up and down at
some frequency f in cycles/second. Any amplitude, and thus any energy of
oscillation is possible. f is smaller for a heavier mass, or larger for a stiffer spring.
All this is well understood from Newtonian mechanics, using Hooke’s Law – the
force on the mass is proportional to the extension or compression of the spring.
However, if Schrödinger’s equation is used, in place of Newtonian mechanics, a
brand new behavior surfaces – you can’t have just any energy, it must have one of
the values (n - ½)hf, where n is an integer 1, 2, 3, . . . For other energy values,
Schrödinger’s equation just yields nonsense. So, there is a comb of possible energy
values, with adjacent ones separated by hf. For everyday springs and masses,
f is of the order of 1 cycle/second; so the energy spacing is so close as to be
indiscernible.
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Even more remarkable, there is a lowest energy (n = 1): ½hf, again so small that no
one would notice – you can’t stop a spring and mass from jiggling. Once again, we
have a quantum number; as in atomic structure, they pop up often. Observe that
for single atomic particles, hf isn’t a negligible energy – it dominates behavior. Soon
after matrix and wave mechanics were formulated, they were applied to the
hydrogen atom; and, after a ton3
of mathematics, Bohr’s model (see 15 Hydrogen
Atom Model) fell right out (see 17 Periodic Table); followed, eventually, by all other
atoms. Yeah, a ton; believe us, or ask any physics grad student.
The importance of the harmonic oscillator solution to physics can’t be overstated.
Think back to the blackbody problem (see 12 Discrete Energy Levels). Planck’s then
ad hoc hypothesis, that that the atoms inside the box could only absorb or radiate
at discrete energies, suddenly acquired a full theoretical justification. Somewhat
later, these energy levels were associated with vibrational and rotational
states of atoms and molecules.
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1
Yeah, a ton; believe us, or ask any physics grad student.

In applying Schrödinger’s equation to particular situations, we make use of any
symmetries that are present. Atoms were known to contain negatively charged
electrons (see 11 Discovery of Electrons) surrounding a positively charged nucleus
(see 14 Discovery of the Nucleus). As it was reasonable to assume spherical
symmetry, Schrödinger’s equation was expressed in spherical coordinates (not to
worry); and it was found that the solutions could be expressed as the product of 3
functions: one purely of the radius from the nucleus, another is a function
depending purely on latitude, and a third purely on longitude. Each function thus
satisfied a separate Schrödinger equation in 1 dimension.
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17: The Periodic Table

The combined latitude and longitude functions always depend on a pair of integers,
l and m. l can have any of the values 0, 1, 2, 3, . . .; while m must lie between –l
and l; i. e. –l, 1 – l, 2 – l, . . ., 0, 1, 2, . . .,l. For any other values of l and m, the
appropriate Schrödinger equation produces nonsense.
In the radial equation, for hydrogen, one must specify the attraction between the
electron and the positive nucleus, and how it depends on the distance between
them. The solution for the energy of the electron then takes the form –K/n2
, where
K is a constant composed of known physical quantities; and n is a positive integer.
The minus sign is to indicate that this is a binding energy, i.e., the energy needed to
free the electron from the nucleus. As with the harmonic oscillator (see 17 Matrix
& Wave Mechanics), l, m, and n are quantum numbers – together, they specify the
state of a hydrogen atom. This expression for the energy exactly conforms to
the successful Bohr model of the hydrogen atom (see 15 Hydrogen Atom
Model), which now had a firmer theoretical basis.
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Another success of this theory came from singly ionized helium. Helium has 2
electrons, and a quite different spectrum. In the early days, only hydrogen was
susceptible to analysis. However, if one electron is somehow removed from a
helium atom, a hydrogen-like atom remains, though with stronger attraction for the
remaining electron. The possible energies are then given by the above formula,
with K replaced by 4K (we are resisting the urge to explain this).
In the complete helium atom, the 2 electrons are identical; so, with the Pauli
exclusion principle, in the ground (lowest energy) state, they must have paired
opposite spins. Moreover, the Schrödinger equation for this system must include
the repulsion between the 2 electrons. With these complications, the equation
must be solved numerically, and yields a completely different set of energy states,
and resulting spectra. You can see this for yourself by comparing the hydrogen
and helium spectra at our Spectroscopy exhibit.
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In the ground state, the spin paired electrons form a closed shell, that has no
tendency to either give up one of its electrons, or accept a third. This is called the n
= 1 shell, an arrangement shared by all more complex atoms. In lithium, with 3
electrons, one of these is forced into a new shell – the n = 2 shell. In the n = 1 shell,
l = m = 0; but in n = 2, l = 0 or 1; so m could be -1, 0, or 1 in the second case; each
possibility permits 2 spin states, for a total of 8. This accounts for elements lithium
through neon, with 10 total electrons, when the n = 2 shell is filled. Further
complications enter when n = 3, which we won’t elaborate. Suffice it to say that the
electronic structure has been elaborated for all the known elements; and we now
have a full understanding of the whole periodic table, first worked out in 1871 by
Dmitry Ivanovich Mendeleyev (1834 – 1907), originally based on chemical
properties. That today’s chemists have to be quantum mechanics should now be
clear.
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Werner Karl Heisenberg (1901 – 1976) was born in Wurzburg, Germany, but moved
to München in 1910, where he received his early education. In 1920 he entered the
University of München. He finished his undergraduate and graduate work in 3
years, and in 1923, presented his doctoral dissertation on turbulence in streams of
fluid.
In his early career Heisenberg was at the forefront of dramatic changes taking place
in quantum mechanics. He studied with three leading quantum theorists at three
major centers of quantum research of that time: German physicists Arnold
Sommerfeld (1868 – 1951) at the University of München, and Max Born (1882 –
1970) (see 19 Born) at the University of Göttingen in 1923; and, from 1924 to 1927,
Danish physicist Niels Bohr (1885 – 1962) (see 15 Bohr) at the Institute for
Theoretical Physics in Kobenhavn.
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17: Heisenberg

Heisenberg's interest in Bohr's model of the planetary atom and his comprehension
of its limitations led him to seek a theoretical basis for a new model. Bohr's concept
had been based on the classical motion of electrons in well-defined orbits around
the nucleus, and the quantum restrictions had been imposed arbitrarily. As a
summary of existing knowledge and as a stimulus to further research, the Bohr
atomic model had succeeded admirably, but new results were becoming more and
more difficult to reconcile with it.
In June 1925, Heisenberg solved a major physical problem — how to account for
the discrete energy states of an anharmonic oscillator. His solution, because it was
analogous to that of a simple planetary atom, launched the program for the
development of the quantum mechanics of atomic systems. Heisenberg published
his results some months later in the Zeitschrift für Physik under the (English)
title “About the Quantum-Theoretical Reinterpretation of Kinetic and
Mechanical Relationships”. Here, he proposed a revision of the basic
concepts of mechanics.
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17: Heisenberg

Heisenberg's treatment of the problem departed from Bohr's as much as Bohr's had
from 19th century tenets. Heisenberg was willing to sacrifice the idea of discrete
particles moving in prescribed paths (neither particles nor paths could be
observed), in exchange for a theory that would deal directly with observations, and
lead to the quantum conditions as consequences of the theory, rather than ad hoc
stipulations. Physical variables were to be represented by arrays of numbers
(matrices); under the influence of Einstein's paper on relativity (1905), he took the
variables to represent not hidden, inaccessible structures but “observable” (i.e.,
measurable) quantities. Born saw that the arrays obeyed the rules of matrix
algebra; he, Pascual Jordan (1902 – 1980), and Heisenberg were able to express the
new theory in terms of this branch of mathematics, and the new quantum theory
became matrix mechanics. Each matrix, usually with an infinite number of rows
and columns, specified the set of possible values for a physical variable, and the
individual terms of a matrix were taken to generate probabilities of occurrences
of states and transitions among states.
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17: Heisenberg

Heisenberg used the new matrix mechanics to interpret the dual spectrum of the
helium atom (that is, the superposed spectra of its two forms, in which the spins of
the two electrons are either parallel or antiparallel), and with it he predicted that
the 2 atom hydrogen molecule should have analogous dual forms. With others, he
also addressed many atomic and molecular spectra, ferromagnetic phenomena,
and electromagnetic behavior. Important alternative forms of the new quantum
theory were proposed in 1926 by Erwin Schrödinger (1887 - 1961) (see 17
Schrödinger) (wave mechanics), and P. A. M. Dirac (1902 – 1984) (see 21 Dirac)
(transformation theory).
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17: Heisenberg
Werner Karl Heisenberg, 1933

In 1927 Heisenberg published the indeterminacy, or uncertainty, principle (see 17
Matrix & Wave Mechanics). Bohr and Heisenberg elaborated a philosophy of
complementarity1
to take into account the new physical variables and an
appropriate measurement process on which each depends. This new conception of
the measurement process in physics emphasized the active role of the scientist,
who, in making measurements, interacts with the observed object, thus causing it
to be revealed, not intrinsically, but as a function of measurement. Many
physicists, including Einstein, Schrödinger, and Louis de Broglie, refused to accept
the philosophy of complementarity.
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17: Heisenberg
1
The idea that things appear to be waves or particles, depending on how you look at them; i.e., that the
measurement process plays an essential role.

From 1927 to 1941 Heisenberg was professor at the University of Leipzig. For the
following 4 years, he was director of the Kaiser Wilhelm (now Max Planck) Institute
for Physics in Berlin. Although he did not publicly oppose the Nazi regime, he was
hostile to its policies. During World War II he worked with Otto Hahn, one of the
discoverers of nuclear fission, on the development of a nuclear reactor. He failed to
develop an effective program for nuclear weapons, probably from want of technical
resources and lack of will to do so. After the war he organized and became director
of the Max Planck Institute for Physics and Astrophysics at Göttingen, moving with
the institute, in 1958, to München; he was also, in 1954, the German representative
for organizing CERN.
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17: Heisenberg

Heisenberg married Elisabeth Schumacher in 1937; they had 7 children. He loved
music in addition to physics and saw a deep affinity between these two interests.
He also wrote philosophical works, believing that new insights into the ancient
problems of Part and Whole, and One and Many would help discovery in
microphysics. Widely acknowledged as one of the seminal thinkers of the 20th
century, Heisenberg was honored with the Max Planck Medal, the Matteucci Medal,
and the Barnard College Medal of Columbia University, in addition to the Nobel
Prize in 1932.
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Wolfgang Pauli (1900 – 1958) was born in Vienna, Austria. He was educated at the
University of München. He taught physics at the Universities of Göttingen (1921-
1922), Kobenhavn (1922-1923), and Hamburg (1923-1928); and was professor of
theoretical physics at the Federal Institute of Technology in Zürich from 1928 until
his death. He also served as visiting professor at the Institute for Advanced Study at
Princeton, New Jersey (1935-1936, 1940-1945, 1949-1950, and 1954).
In 1925 Pauli defined the exclusion principle, which states that no two electrons can
occupy the same quantum state simultaneously in an atom. 4 quantum numbers —
principle, angular momentum, magnetic, and spin quantum numbers — define the
quantum state of an electron. Pauli’s hypothesis in 1930 of the existence of the
neutrino, a subatomic particle, was a fundamental contribution to the development
of meson theory. He was awarded the 1945 Nobel Prize in physics.
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Erwin Schrödinger (1887 – 1961) was born in Vienna, Austria. He entered the
University of Vienna in 1906 and obtained his doctorate in 1910, after which he
accepted a research post at the university's Second Physics Institute. He saw
military service in World War I and then went to the University of Zürich in 1921,
where he remained for the next 6 years. There, in a six-month period in 1926, at
the age of 39, he produced the papers that gave the foundations of quantum wave
mechanics. In those papers he described his partial differential equation that is the
basic equation of quantum mechanics and bears the same relation to the
mechanics of the atom as Newton's equations of motion bear to planetary
astronomy. Adopting a proposal made by Louis de Broglie (see 16 De Broglie) in
1923, that particles of matter often act like waves, Schrödinger introduced a theory
describing the behavior of such a system by a wave equation that is now known
as the Schrödinger equation.
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17: Schrödinger

The solutions to this equation, unlike solutions to Newton's equations, are wave
functions that can only be related to the probable occurrence of physical events.
The deterministic sequence of events of the planetary orbits of Newton is, in
quantum mechanics, replaced by the more abstract notion of probability. This
aspect of the quantum theory made Schrödinger and several other physicists
profoundly unhappy, and he devoted much of his later life to formulating
philosophical objections to the generally accepted (Born) interpretation of the
theory that he had done so much to create.
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17: Schrödinger
Erwin Schrödinger

In 1927 Schrödinger accepted an invitation to succeed Max Planck, the inventor of
the quantum hypothesis, at the University of Berlin; and he joined an extremely
distinguished faculty that included Albert Einstein (see 13 Einstein). He remained at
the university until 1933, at which time he decided that he could no longer live in a
country in which the persecution of Jews had become a national policy. He then
began a 7 year odyssey that took him to Austria, Great Britain, Belgium, the
Pontifical Academy of Science in Rome, and, finally in 1940, the Dublin Institute for
Advanced Studies, founded under the influence of Premier Eamon de Valera, who
had been a mathematician before turning to politics. Schrödinger remained in
Ireland for the next 15 years, doing research both in physics and in the philosophy
and history of science. During this period he wrote What Is Life? (1944), an attempt
to show how quantum physics can be used to explain the stability of genetic
structure. Although much of what Schrödinger had to say in this book has
been modified and amplified by later developments in molecular biology, his
book remains one of the most useful and profound introductions to the
subject. In 1956 Schrödinger retired and returned to Vienna as professor
emeritus at the university.
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17: Schrödinger

Of all of the physicists of his generation, Schrödinger stands out because of his
extraordinary intellectual versatility. He was at home in the philosophy and
literature of all of the Western languages, and his popular scientific writing in
English, which he had learned as a child, is among the best of its kind. His study of
ancient Greek science and philosophy, summarized in his Nature and the Greeks
(1954), gave him both an admiration for the Greek invention of the scientific view
of the world, and skepticism toward the relevance of science as a unique tool with
which to unravel the ultimate mysteries of human existence. Schrödinger's own
metaphysical outlook, as expressed in his last book, Meine Weltansicht 1961; (My
View of the World), closely paralleled the mysticism of the Vedānta. Because of his
exceptional gifts, Schrödinger was able to make significant contributions to nearly
all branches of science and philosophy, an almost unique accomplishment at a time
when the trend was toward increasing technical specialization.
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In 1925, Clinton Joseph Davisson (1881 - 1958), then at the Western Electric Co.,
was bombarding a nickel target with electrons. A laboratory accident caused the
nickel to be oxidized. Prolonged heating, to drive off the oxygen, caused the
surface to crystallize; but he went ahead with his measurements. To his surprise,
the reflection pattern had completely changed, as though he had been using x-rays
instead of electrons.
The next year, he heard about Louis de Broglie’s idea (see 16 Electrons are Waves)
that electrons had wave properties; and with Lester Halbert Germer (1896 - 1991),
in 1927 demonstrated that the pattern was truly diffraction, caused by the electron
waves bouncing off the regular array of atoms in the nickel crystals. Davisson
shared the 1937 Nobel Prize in physics with J. J. Thomson (see 11 Thomson).
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Davisson (left) with Lester Germer, 1927

To get a better idea of what’s going on in quantum mechanics, we need to discuss
Schrödinger’s equation in more (non-mathematical) detail. To work with it, one
ladles in the particulars of the problem at hand, and the initial conditions of
anything that moves. One then solves this equation, numerically in most cases, to
get something called the “wave function”, Ψ, a creature that has a value at every
point in space and time, like temperature in this respect.
Initially, Ψ was used to compute the possible energy levels in atoms, and other
simple atomic systems. When the solutions for energy are discrete, values between
solutions simply don’t satisfy Schrödinger’s equation, and don’t occur. As photons
are emitted or absorbed from an atom, when an electron jumps from one level to
another, they can’t have just any wavelength or color, accounting for the discrete
nature of atomic spectra.
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19: Statistical Interpretation

Around 1926, Max Born (1882 - 1970) brought new meaning to Ψ. Noting that, if Ψ
is a solution to Schrödinger’s equation, then so is Ψ times any number; so, if Ψ is to
have any physical meaning, you needed to choose the number carefully. Born’s
insight was that, at any given time and place, Ψ times itself is the probability density
of finding a particle there. If, at a given time, you sum this quantity over a given
region, you get the probability of finding the particle in that region1
. The correct
number to multiply by is obtained by insisting that the probability that the particle
is somewhere is 1.
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1
Integration, for the mathematically astute.

Let’s contrast this with classical physics; i.e., Newton’s laws and the Maxwell
equations. There, solutions of the equations yield the trajectories of the particles,
or waves, or whatever. Here, what you fish out of Schrödinger’s equation is the
likelihood of finding your electron, or photon, or whatever, in a given region. A
crucial part of this interpretation is that you look there with some sort of sensor or
detector. If you detect something, Ψ collapses to what you saw; whereas, without
a measurement, the whatever goes about its business undisturbed, until you
somehow get in its way.
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To illustrate this behavior, Schrödinger, in 1935, and in a lighter mood, proposed
the following thought experiment, now known as Schrödinger’s Cat. Suppose you
stuff a cat into a box, also containing a bit of radioactive material and a detector of
its radiation. If the detector trips, it releases a lethal gas into the box. After some
time has passed, you inquire if the cat is alive or dead. Classically, you can compute
that the probability is so and so that the cat is still alive; but quantum mechanically,
the cat is both alive and dead, until you lift the lid of the box, when the cat’s overall
wave function collapses to one state or the other. Confused? Schrödinger himself
thought this was blather; but physicists have argued about it ever since. One way
to look at this is that, by opening the box, you (maybe) murdered the cat. Another
is that the detector, if it measures something, collapses the wave function, and kills
the cat, thus absolving you.
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A completely new wrinkle in this argument was put forward around 1956 by Hugh
Everett III (1930 – 1982) in his Ph. D. thesis. In his view, when you lifted the lid on
the box, the universe spilt in 2, one with a live cat, and the other with it dead. That
the split might have occurred earlier, when the detector saw a particle (or didn’t) is
merely a quibble; Everett’s interpretation is entirely consistent with Schrödinger’s
Equation. If the idea of zillions and zillions of universes sticks in your craw; the
older idea that the wave function, defined over all space and time, suddenly
collapses when you opened the box, ain’t easy to swallow either.
Schrödinger's Cat

Now, let’s return to Thomas Young’s experiment in 4 Interference, that “proved”
that light is waves. Referring to the illustration there, the final screen is our
detector, say a photographic plate. Classically, if the distances from a point C on
the plate to the 2 slits differ by an integral number of wavelengths, there is
constructive interference, and C is a point of maximum intensity. Similarly, if the 2
distances differ by an odd number of half wavelengths, as at D, the interference is
destructive; and the waves cancel out.
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Quantum mechanically, the wave function at any point on the plate is the sum of
the wave functions of whatever comes through the 2 slits. So, the probability of
finding photons at D vanishes, and is maximum at C, from which you might
conclude that light is particles (photons). But, what you actually measure is
whether each grain on the plate develops; and all Ψ actually tells you is the
probability of that happening, or not, at each point of the plate. So, it makes no
difference whether light is regarded as waves or particles.

That the latter interpretation makes more sense has been shown by reducing the
intensity of the light source till only a few photons fall on the plate. Then, the
interference pattern builds up slowly and chaotically, contrary to the classical
picture, where, below some minimum intensity, the photographic grains wouldn’t
develop at all. The Davisson – Germer experiment looks much the same (see 18
Electron Diffraction). There, the regular arrangement of atoms in the nickel crystal
act as a diffraction grating, amounting to a Thomas Young experiment, with a huge
number of slits. And whether you regard electrons as waves or particles again
makes no difference. This then is the solution to the wave – particle controversy;
and if you don’t find it satisfactory, you’re not the first.
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One more strange behavior to come out of the statistical interpretation is that of
“quantum tunneling”. To introduce this, consider a classical analogy. Imagine a
volcano crater, with smooth insides. Let go of a frictionless ball somewhere up the
inside. Classically, the ball will slide back and forth inside, forever. However, if you
solve Schrödinger’s equation for the ball, the resulting wave function will be non-
zero everywhere below the initial height, including outside the crater. So, the
statistical interpretation tells us that the ball can’t be within the crater walls; but
there’s some chance that it could leak through the wall, and fall down the outside
of the crater. For everyday balls and volcanos, you’d have to wait a very long time
for the ball to leak through; but, on the atomic scale, it happens all the time, and is
the basis for several devices we’ll discuss in 23 Solid State Theory & Practice, and of
radioactivity, as we’ll see in 26 Nuclear Physics.
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Max Born (1882 – 1970) was born in Breslau, Germany (now Wroclaw, Poland).
Born earned a Ph.D. in physics from the University of Göttingen in 1907 and
eventually began teaching there. In 1921, the year he became professor of
theoretical physics at Göttingen, Born produced a very precise definition of quantity
of heat, the most satisfactory mathematical statement of the first law of
thermodynamics (see 6 Thermodynamics & Statistical Mechanics). In 1926, after
his student Werner Heisenberg (see 17 Heisenberg) had formulated the first laws of
a new quantum theory, Born collaborated with him to develop the mathematical
formulation that would adequately describe it. Somewhat later, when Erwin
Schrödinger (see 17 Schrödinger) put forward his quantum mechanical wave
equation, Born showed that the solution of the equation has a statistical meaning
of proved to be of fundamental importance in the new theory of quantum
physical significance.
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19: Born

In 1933 Born fled the Nazis and became the Stokes Lecturer at the University of
Cambridge. He was elected to the Tait chair of natural philosophy at the University
of Edinburgh in 1936, becoming a British subject in 1939. After his retirement in
1953 Born returned to Göttingen. He won the Nobel Prize for Physics in 1954, with
Walther Bothe (1891 – 1957), also from Germany.
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19: Born
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For the most part, quantum mechanics, as formulated by Heisenberg and
Schrödinger, ignores relativity. Attempts to correct this will be taken up in 21
Quantum Electrodynamics and 21 Dirac. Here, a short review will be undertaken as
preparation. For this, we must return to the great successes of the wave theory of
light in the 19th century (see 7 Electromagnetic Theory). One mystery that
confounded physicists was, if light is waves, then waves of what? Naming this stuff
the “luminiferous aether” explained nothing. Many wondrous properties were
ascribed to it; but one stood out – since we could see distant stars, it must pervade
all space.
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20: Special Relativity

In 1887, Albert Abraham Michelson (1852 - 1931), and Edward Williams Morley
(1838 – 1923) set out to measure the earth’s motion through it. They used an
interferometer, a previous invention of Michelson, and shown in the figure. The
mirrors are adjusted so that, with no motion, the 2 beams will arrive at the detector
(the eyes) in phase, and will constructively interfere. However, if the earth is in
motion to the right about the sun, and thus plowing through the ether, the purely
horizontal beam as shown should arrive at the detector a trifle late, thus altering
the interference pattern. The whole apparatus was about a meter in diameter,
mounted on a stone block, and floated in mercury.

The experimenters slowly turned the block 90 degrees to interchange the paths.
Their calculation of the expected phase shift was straightforward, if tedious; but,
there was no change in the pattern. Profound shock. Evidently, the earth wasn’t
moving through the ether. The experiment has been performed many times
since, by many others, with the same result; and physics slowly crumbled.

In 1895, Hendrik Antoon Lorentz (1853 - 1928), found a transformation that
contracted any object in its direction of motion, by just enough to account for the
Michelson – Morley results. Unfortunately, he had no physical reason why this
should happen. So matters stood until 1905, when Albert Einstein (1879 – 1955)
(there he is again) (see 13 Einstein) published what is now called the Special Theory
of Relativity. It consisted of 2 postulates, from which all kinds of wonders flowed:
1) The laws of physics are the same in any set of unaccelerated coordinate systems.
2) The speed of light, c = 299,776 kilometers/second, is the same in all such
systems.
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An example may help to confuse you. Fix a light source in a train station. With
instruments in the station, measure the speed of this light. You get c. Now suppose
a train is approaching this station at some hellish speed v. Your light passes
lengthwise through a window in the approaching train; and its speed is measured
there by your buddy Joe, with similar instruments. He gets c, not c + v, or anything
in between. Believe us, this experiment has been performed many, many times, in
many ways, always with this result.
There’s a lot more to this. If Joe compares his clock on the train to yours in the
station, he finds his running faster. And if Joe holds up a ruler, to you it appears to
have shortened in the down track direction. All this is consistent with the Lorentz
transformation, which Einstein was able to derive from his postulates above. The
Michelson – Morley results were explained; the luminiferous aether was dead,
and physics was changed forever. Of course, the question of what light was
a wave of remained; but you saw that resolved in 19 Statistical Interpretation,
sort of.
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Another strange effect is that Joe’s mass, as you somehow measure it in the station,
has increased; but if Joe steps on a scale in the train, he sees no change. In special
relativity, we say that his kinetic energy (see 5 Conservation of Energy) due to his
speed v is the source of the added mass, according to Einstein’s ubiquitous formula
E = mc2
. (Right, we promised no mathematics; but as you must have seen this
before, you may regard this formula as purely ornamental.)
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When large energies are released in nuclear reactions, this is the conversion factor.
The 19th century idea of the conservation of energy had now been broadened to
the 20th century notion of the conservation of mass-energy. Another broadening is
to regard time as a fourth dimension, on the same footing as the 3 spatial
dimensions; they are together called “spacetime”. To make time look like space,
you multiply an interval t in seconds by the speed of light c in meters per second, to
get a distance in meters.
Einstein went on to expand all this to general relativity, allowing for accelerated
motion and gravity in 1916; but, as it hasn’t yet been consolidated with quantum
mechanics, we won’t delve further.
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Following the Born interpretation of the wave function as leading to the probability
density of finding something somewhere at any given time (see 19 Statistical
Interpretation), quantum mechanics scored many successes in illuminating many
areas of physics. Still, there were several limitations to the theory. The first major
stretch was a new theory of the electron, introduced in 1928, incorporating special
relativity by Paul Adrien Maurice Dirac (1902 – 1984) (see 21 Dirac). One success of
this theory was to predict the existence of the positron, or positive electron, found
in 1932 in cosmic ray emulsions by Carl David Anderson (1905 – 1991). Dirac was a
major player in the development of quantum mechanics, and with Schrödinger, got
the Nobel Prize in physics in 1936.
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21: Quantum Electrodynamics
Cloud chamber photograph by C.D. Anderson of the first
positron ever identified. A 6 mm lead plate separates
the upper half of the chamber from the lower half. The
positron must have come from below since the upper
track is bent more strongly in the magnetic field
indicating a lower energy.

Besides relativity, the Schrödinger equation can’t handle interactions between
photons and electrons, or the creation or annihilation of particles, often arising in
nuclear interactions. Sometimes, it produced divergent or infinite calculations. A
wider theory, called “quantum electrodynamics”, or QED for short, was contrived
independently by Richard Phillips Feynman (1918 – 1988), Julian Seymour
Schwinger (1918 – 1944), and Shin’ichirō Tomonaga1
(1906 – 1979) (see 21
Feynman et. al.) Each version addressed all these problems; and they jointly
received the 1965 Nobel Prize in physics.
1
In Japan, where the family name is first, this would be Tomonaga Shinichir (or Sin-itiro).
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The impetus for QED was mainly because tiny discrepancies arose between
increasingly accurate spectral measurements of atomic hydrogen, and the
corresponding predictions of matrix or wave mechanics (see 17 Matrix & Wave
Mechanics). Dirac’s approach did somewhat better; but even there, certain
calculations tended to blow up, rather than converge to a finite answer. The QED
results now agree with observations at around 10 decimals accuracy; although the
calculations are extraordinarily difficult, and require enormous amounts of
computer time.
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In the Feynman version, probably the most popular, due to his excellence as an
expositor2
, if an electron is emitted at A, and is detected at B, at specific times, one
must consider all possible paths, including the direct path from A to B, and all sorts
of drunken alternatives. The simplest such path has the electron emitting a
“virtual” photon at some point C in spacetime3
, which is reabsorbed at some
point D in spacetime. The photon is virtual because you can’t actually
observe it.
2
For a non-mathematical rendition, see QED, the strange theory of light and matter, R. P. Feynman,
Princeton University Press, 1985; with a new introduction, 2006.
3
There’s the special relativity, see 20 Special Relativity.

To start the calculations, you assume the electron goes directly from A to B, without
interruption. The first correction consists of averaging over all points C and D in
spacetime, and runs around 1% of the starting calculation4
. A higher order
correction comes from considering a second photon, emitted and absorbed at some
points E and F; or somewhere on the photon’s path it creates a virtual electron-
positron pair, which annihilates to produce a new photon. As Feynman pointed
out, if E precedes F, it’s perfectly consistent to regard the virtual positron as
originating at F, and going backwards in time to annihilate with the virtual electron
at E.
4
The reduction factor is actually 1/α = 137.036, where α is called the fine structure constant.

After averaging over everywhere and when, this correction is down by another 1%
from the first. Each additional correction involves all the ways you can introduce
another pair of interactions, and is less likely by about the same factor. To get to 10
decimal accuracy, physicists have had to consider zillions of scenarios, each
averaged over all spacetime. Feynman himself put in a lot of effort looking for ways
to cut the work, as have hundreds of grad students since. As they have found
improvements, and computer speeds have grown, experimental physicists have had
to strain to improve the measurements, an ongoing battle to get or stay ahead.

Perhaps the most important application of QED is in the structure of atoms. In
ordinary garden variety quantum mechanics, we say that electrons and nuclei have
electric fields caused by their charge. These fields are derivable from a “potential
function” which depends only on the distance from a charge; and the potential
function enters directly into Schrödinger’s equation. Classically, the fields cause
forces between the electrons and nuclei, and amongst the electrons; but forces
don’t appear explicitly in Schrödinger’s equation. In QED, all this goes away, to be
replaced by virtual photons running back and forth continually between all the
charged particles. You will encounter these ideas again in 26 Nuclear Physics, and
27 Quantum Chromodynamics, and finally in 28 Theory of Everything.
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Paul Adrien Maurice Dirac (1902 – 1984) was born in Bristol, England. His mother
was British and his father Swiss. Dirac's childhood was unhappy — his father
intimidated the children, both at home and at school where he taught French, by
meticulous and oppressive discipline. Dirac grew up an introvert, spoke only when
spoken to, and used words very sparingly — though with utmost precision. In later
life, Dirac was proverbial for his lack of social and emotional skills, and his lack of
small talk. He preferred solitary thought and long walks, and had few, though very
close, friends. Dirac showed from early on extraordinary mathematical abilities, but
few other interests. His physics papers and books, however, are literary gems.
21: Dirac
Paul Adrien Maurice Dirac

On his father's wish, Dirac studied electrical engineering at the University of Bristol
(1918 – 21). Not having found employment upon graduation, he took two more
years of applied mathematics. Albert Einstein's theory of relativity had become
famous after 1919. Fascinated with the technical aspect of relativity, he mastered it
on his own. Following the advice of his mathematics professors, and with the help
of a fellowship, he entered the University of Cambridge as a research student in
1923. His adviser, Ralph Fowler, was then the only professor in Cambridge at home
with the new quantum theory being developed in Germany and Denmark.
21: Dirac

In August 1925, through Fowler, Dirac received proofs of an unpublished paper by
Werner Heisenberg (see 17 Matrix & Wave Mechanics) that initiated the
revolutionary transition from the Bohr atomic model to the new quantum
mechanics. In a series of papers, and his 1926 Ph.D. thesis, Dirac further developed
Heisenberg's ideas. Dirac's accomplishment was more general in form but similar in
results to matrix mechanics, another early version of quantum mechanics. In the fall
of 1926 Dirac and, independently, Pascual Jordan combined the matrix approach
with the powerful methods of Schrödinger's wave mechanics (again see 17 Matrix &
Wave Mechanics), and Born's statistical interpretation (see 19 Statistical
Interpretation) into a general scheme — transformation theory — that was the first
complete mathematical formalism of quantum mechanics. Along the way, Dirac
also developed the Fermi-Dirac statistics (which had been suggested somewhat
earlier by Enrico Fermi).
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21: Dirac

Satisfied with the interpretation that the fundamental laws governing microscopic
particles are probabilistic, or that “nature makes a choice”, Dirac declared quantum
mechanics complete and turned his main attention to relativistic quantum theory.
His 1927 quantum theory of radiation is often regarded as the true beginning of
quantum electrodynamics (see 21 Quantum Electrodynamics). In it Dirac developed
methods of quantizing electromagnetic waves and invented the so-called second
quantization — a way to transform the description of a single quantum particle into
a formalism of the system of many such particles. In 1928 he published what may
be his greatest single accomplishment — the relativistic wave equation for the
electron. In order to satisfy the condition of relativistic invariance (i.e., treating
space and time coordinates on the same footing), the Dirac equation required a
combination of 4 wave functions and relatively new mathematical quantities
known as spinors. As an added bonus, the equation described electron spin
— a fundamental but not earlier explained feature of quantum particles
(see 17 Matrix & Wave Mechanics).
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21: Dirac

From the beginning, Dirac was aware that his spectacular achievement also suffered
grave problems: it had an extra set of solutions that made no physical sense, as it
corresponded to negative values of energy. In 1930 Dirac suggested a change in
perspective to consider unoccupied vacancies in the sea of negative energy
electrons as positively charged “holes”. By suggesting that such “holes” could be
identified with protons, he hoped to produce a unified theory of matter, as
electrons and protons were then the only known elementary particles. Others
proved that a “hole” must have the same mass as the electron, whereas the proton
is far heavier. This led Dirac to admit in 1931 that his theory, if true, implied the
existence of a new kind of particle, unknown to experimental physics, having the
same mass and opposite charge to an electron. One year later, to the astonishment
of physicists, this particle — the anti-electron, or positron — was
accidentally discovered in cosmic rays by Carl Anderson (1905 – 1991) of the
United States.
21: Dirac

An apparent difficulty of the Dirac equation thus turned into an unexpected
triumph and one of the main reasons for Dirac's being awarded the 1933 Nobel
Prize for Physics along with the Austrian physicist Erwin Schrödinger.
In his later work, Dirac continued making important improvements and
clarifications in the logical and mathematical presentation of quantum mechanics,
in particular through his influential textbook The Principles of Quantum Mechanics
(1930, with three subsequent major revisions). The professional terminology of
modern theoretical physics owes much to Dirac, including (for the few physicists
among you) the names and mathematical notations fermion, boson, observable,
commutator, eigenfunction, delta-function, ℏ (for h/2π, where h is Planck's
constant), and the bra-ket vector notation.
21: Dirac

Relativistic quantum theory seemed incomplete to him. In the 1930s quantum
electrodynamics encountered serious problems; in particular, infinite results
appeared in various mathematical calculations. Dirac was even more concerned
with the formal difficulty that relativistic invariance did not follow directly from the
main equations, which treated time and space coordinates separately. Searching
for remedies, Dirac in 1932–33 introduced the “many-times formulation” and the
quantum analog for the principle of least action, later developed by Richard
Feynman (see 21 Feynman et. al.).
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21: Dirac

These concepts, and also Dirac's idea of vacuum polarization (1934), helped a new
generation of theorists after World War II invent ways of subtracting infinities from
one another in their calculations so that predictions for physically observable
results in quantum electrodynamics would always be finite quantities. Although
very effective in practical calculations, these “renormalization” techniques
remained, in Dirac's view, clever tricks rather than a principled solution to a
fundamental problem. He hoped for a revolutionary change in basic principles that
would eventually bring the theory to a degree of logical consistency comparable to
what had been achieved in nonrelativistic quantum mechanics. Although Dirac
probably contributed more to quantum electrodynamics than any other physicist,
he died dissatisfied with his own brainchild.
21: Dirac

Dirac taught at Cambridge after receiving his doctorate there, and in 1932 he was
appointed Lucasian Professor of Mathematics, the chair once held by Isaac Newton.
Although Dirac had few research students, he was very active in the research
community through his participation in international seminars. Unlike many
physicists of his generation and expertise, Dirac did not switch to nuclear physics
and only marginally participated in the development of the atomic bomb during
World War II. In 1937 he married Margit Balasz (née Wigner; sister of Hungarian
physicist Eugene Wigner). Dirac retired from Cambridge in 1969 and, after various
visiting appointments, held a professorship at Florida State University, Tallahassee,
from 1971 until his death.
21: Dirac
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Richard Phillips Feynman (1918 – 1988) was born in New York City, descendant of
Russian and Polish Jews who had immigrated to the United States late in the 19th
century. He studied physics at MIT, where his undergraduate thesis (1939)
proposed an original and enduring approach to calculating forces in molecules.
Feynman received his doctorate at Princeton University in 1942. There, with his
adviser, John Archibald Wheeler (1911 - 2008), he developed an approach to
quantum mechanics governed by the principle of least action, which replaced the
wave-oriented electromagnetic picture developed by James Clerk Maxwell with one
based entirely on particle interactions mapped in space and time. In effect,
Feynman's method calculated the probabilities of all the possible paths a particle
could take in going from one point to another.
21: Feynman Et. Al.
Richard Feynman

During World War II Feynman was recruited as a staff member of the U.S. atomic
bomb project at Princeton University (1941 – 42), and then at Los Alamos (1943 –
45). There, he became the youngest group leader in the theoretical division of the
Manhattan Project. With the division head, Hans Bethe (see 24 Astrophysics), he
devised the formula for predicting the energy yield of a nuclear explosive. He
observed the first atomic bomb detonation on July 16, 1945, near Alamogordo, and,
though his initial reaction was euphoric, he later felt anxiety about the force he and
his colleagues had helped unleash on the world.
At war's end Feynman became an associate professor at Cornell University (1945 –
50) and returned to studying the fundamental issues of quantum electrodynamics.
In 1950, he became professor of theoretical physics at the California Institute of
Technology (Caltech) for the rest of his career.
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21: Feynman Et. Al.

In the early 1950s Feynman provided a quantum mechanical explanation for the
theory of superfluidity — the strange, frictionless behavior of liquid helium at
temperatures near absolute zero. In 1958 he and Murray Gell-Mann (see 27 Eight-
Fold Way) devised a theory that accounted for most of the phenomena associated
with the weak force, the force at work in radioactive decay. Their theory, which
turns on the asymmetrical “handedness” of particle spin, proved particularly fruitful
in modern particle physics.
Finally, in 1968, while working with experimenters at the Stanford Linear
Accelerator, Feynman invented a theory of “partons”, or hypothetical hard particles
that helped lead to the modern understanding of quarks.
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21: Feynman Et. Al.

Feynman's stature among physicists transcended the sum of even his sizable
contributions to the field. His bold and colorful personality, unencumbered by false
dignity or notions of excessive self-importance, seemed to announce: “Here is an
unconventional mind”. His purely intellectual reputation became part of the
scenery of modern science. Feynman diagrams, Feynman integrals, and Feynman
rules joined Feynman stories in the everyday conversation of physicists. Feynman's
lectures at Caltech evolved into the books Quantum Electrodynamics (1961) and
The Theory of Fundamental Processes (1961). In 1961 he began reorganizing and
teaching the introductory physics course at Caltech; the result, published as The
Feynman Lectures on Physics, 3 vol. (1963 – 65), became a classic textbook.
Feynman's views on quantum mechanics, scientific method, the relations between
science and religion, and the role of beauty and uncertainty in scientific
knowledge are expressed in two models of science writing, again distilled
from lectures: The Character of Physical Law (1965), and QED: The strange
theory of light and matter (1985).
21: Feynman Et. Al.

Julian Seymour Schwinger (1918 – 1994), another New Yorker, received his Ph. D.
from Columbia University at age 21. From 1939 to 1941, he worked under J. Robert
Oppenheimer (1904 – 1967) at the University of California, Berkeley. In 1945,
Schwinger joined the faculty of Harvard University, where in 1947 he became one
of the youngest full professors in that university's history. It was at Harvard that he
began working to correct P. A. M. Dirac's (see 21 Dirac) mathematical account of
the relations between charged particles and electromagnetic fields. Schwinger's
equations not only saved Dirac's theory, but served to unite electromagnetic theory
with quantum mechanics to form the new field of quantum electrodynamics.
Schwinger completed his work unaware that Feynman and Tomonaga were
independently working on the same problem. From 1972 until his death he was
professor of physics at the University of California, Los Angeles.
21: Feynman Et. Al.
Julian Seymour Schwinger Tomonaga Shin'ichirō

Tomonaga Shin'ichirō (1906 – 1979) was born in Kyōto, Japan. Tomonaga became
professor of physics at Bunrika University (later Tokyo University of Education) in
1941, the year he began his investigations of the problems of quantum
electrodynamics. World War II isolated him from Western scientists, but in 1943,
he completed and published his research. Tomonaga's theoretical work made
quantum electrodynamics consistent with the theory of special relativity. It was
only after the war, in 1947, that his work came to the attention of the West, at
about the same time that Feynman and Schwinger published the results of their
research. It was found that all three had achieved essentially the same result from
different approaches, and had resolved the inconsistencies of the old theory
without making any drastic changes.
Tomonaga was president of the Tokyo University of Education from 1956 to 1962,
and the following year he was named Chairman of the Japan Science Council.
Throughout his life Tomonaga actively campaigned against the spread of
nuclear weapons and urged that resources be spent on the peaceful use of
nuclear energy.
21: Feynman Et. Al.
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Modern chemistry can be said to have begun with Linus Carl Pauling (1901 – 1994),
and his quantum mechanical description of the chemical bond. Polar bonds had
long been known, in which one atom donates an electron to another. The donor is
then positively charged, and is attracted to the negatively charged acceptor. An
example is sodium chloride (table salt), in which a sodium atom donates an electron
to a chlorine atom, when they then stick together. When salt is dissolved in water,
some of the charged atoms, or “ions” as they are called, float freely among the
water molecules, and may bond to other ions, if present.
Alone, a sodium atom has 1 electron in its outer shell (see 17 Periodic Table), while
chlorine has 7. If the chlorine atom steals the sodium atom’s outer electron, both
will have complete outer shells, a quantum mechanically desirable arrangement.
Pauling worked out the details, calculated the strength of the bond (the amount
of energy needed to break the bond), and published his results in 1931 in the
now classic book, “The Nature of the Chemical Bond”.
22: Modern Chemistry and Molecular Biology

Another arrangement is the “covalent bond”. An example is the chlorine molecule,
in which 2 chlorine atoms share a pair of outer electrons of opposite spin. They
then have complete outer shells, sort of, and will cling together to maintain this
condition. Again, Pauling worked out the quantum details, which you may also find
in his book. Many diatomic gasses have this structure; e.g., hydrogen, in which both
electrons surround the 2 nuclei; oxygen with 6 outer electrons, shares 2; and
nitrogen with 5 outer electrons, shares 3. Pauling received the 1954 Nobel Prize in
chemistry for his many contributions.
Chlorine molecule

Chemists soon built upon this pioneering work by analyzing molecules of increasing
complexity. Pauling himself tackled benzene, (C6H6) first examined by Michael
Faraday (see 7 Faraday). Its ring structure was first proposed by Friedrich August
Kekulé von Stradonitz (1829 – 1896) in 1865, but made no sense until Linus Pauling
(1901 – 1994) applied the emerging quantum mechanics to it. Kekulé had found
that benzene has 6 carbon atoms in a ring, with a hydrogen atom attached to each1
.
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1
Kekulé said that he saw the benzene molecule in a dream as a snake chasing its tail.
Kekulé Structures

What Pauling found was that the carbons are connected by alternating single and
double covalent bonds, with 1 hydrogen atom attached covalently to each carbon.
Do these bonds chase each other around the ring; if so which way do they go; or do
they simply switch between single and double? The question is meaningless from
the quantum viewpoint, as the actual behavior is unobservable without interfering
with the molecule. Pauling called this a resonant molecule, worked out the
quantum properties, and showed that many molecules behave similarly.
Thousands of very important compounds are in use today, based on benzene
rings and their derivatives.
The growing understanding of chemical structure gradually led into “organic
chemistry” – the study of compounds found in living things. These had long been
thought to possess some special property, unique to life; but as more and more
were synthesized, it was recognized that there was no special property of
organic compounds, other than that they were generally built on chains and
rings based mostly on carbon.

One enormous class is the “proteins”, made up of chains of “amino acids”. Each
amino acid has a central sequence C-C-N, where C is for carbon, and N is for
nitrogen. All sorts of side groups can be attached to the central C; but plants and
animals use only 20 of these. Different proteins are distinguished by what amino
acids appear in the chain, in what order, and how they are folded up into 3
dimensional structures. Pauling recognized that many proteins contained a
subgroup, consisting of a spiral structure of amino acids he called the a-helix. He
was able to find the complete structure, including how many amino acids are in
each turn of the helix. Today, many thousands of proteins are known in detail; and
their biological functions are gradually coming to light.
A-helix model

Another huge class are the nucleic acids, RNA and DNA2
. Each is made of a long
string of “nucleotides” - adenine, guanine, cytosine, and uracil in RNA, with thymine
replacing uracil in DNA. Each group of 3 sequential nucleotides is called a “codon”;
there are 64 codons (4×4×4), each of which codes for one of the amino acids, plus
various punctuation marks. Every protein is associated with one or both nucleic
acids, which specify its synthesis in all living things. This story (molecular biology)
fills libraries; so we’ll only note that, without quantum mechanics, we would
understand very little of it.
2
Ribonucleic acid and deoxyribonucleic acid.
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Essentially everything we have today in the way of electronic and optical devices
depends on modern developments in quantum mechanics. We’ll delve into a few.
First, crystals. Most solid materials have a structure consisting of a regular 3
dimensional array of nuclei, each surrounded by a set of closed shells of electrons.
The noble gasses (helium, neon, argon, etc.), cooled to solids, will bind together due
purely to weak forces from slight distortions of the outer electron clouds. Other
atoms or compounds may crystallize due to polar or covalent bonds between the
outer shell electrons (see 22 Modern Chemistry).
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23: Solid State Theory and Practice

Quantum mechanics requires that, in any large number of atoms in a crystalline
array, the energies allowed for the outer electrons fall into a set of bands – none of
these electrons can have an energy in the gaps between the bands. Each band
contains a whole lot of levels, which, by the exclusion principle, can each be
populated by 2 electrons at most. At low temperatures, the electrons tend to
populate the lowest energy levels; which means that the lower bands will be
completely filled; the upper bands will be empty; and some intermediate energy
band will be partly filled. At absolute zero temperature, the filled levels end at an
energy known as the “Fermi level”, after Enrico Fermi (1901 – 1954). As the
temperature is raised, some electrons will populate higher levels. At higher
temperatures, a few may jump up to the next band, each leaving behind a “hole”,
which acts rather like a positive charge.

Above some number of “valence bands” there is generally a “conduction band”, in
which electrons are free to roam about the crystal. If the gap below the conduction
band is wide, very few electrons will be free; and the material is an “insulator”, i.e.,
a poor conductor of heat and electricity. If that gap is narrow, lots of electrons will
be free to roam; and the material is a “conductor”, or a good conductor of heat and
electricity. For example, silver is not only the best metallic conductor of electricity;
but the handles of silver teapots are often too hot to touch.
Now, connect a battery across 2 points of a material, and the free electrons will
move toward the “plus” terminal, and the holes toward the “minus” terminal.
When a hole moves, to the right say, a valence electron just to the right falls into it,
leaving a new hole just to the right, etc. Books are written about the relative
mobilities of electrons and holes; but we’re not going to get into it.

Twixt conductors and insulators, there are intermediate materials called
semiconductors. Typically, the intrinsic conductivity is low; but, with the addition of
a tiny impurity, will climb enormously. Impurities with 1, 2, or 3 outer electrons are
called “donor” materials – their electrons will occupy levels just below the main
material conduction band, when only a little thermal excitation will free them.
“Acceptor” impurities, with 5, 6, or 7 outer electrons, will sit just above the highest
valence band, sucking in electrons, and causing holes to appear.
The former are called “n” type semiconductors, as the primary current carriers are
electrons (negative); while the latter are “p” type semiconductors, as the primary
current carriers are holes (positive).
Diagram of “N” and “P” Layer semiconductors

The most important such material is pure silicon, partly because it’s one of the
earth’s commonest elements, partly because it’s easily purified, and partly because
it has a convenient gap width. However, several other materials have properties
that are useful in certain applications; e.g., germanium, gallium arsenide, and
indium antimonide.
The earliest application of these ideas was the “junction diode”. Take a thin slab of
silicon, expose it to a hot donor vapor on one side, and acceptor vapor on the other,
and the 2 will diffuse into the silicon, causing a junction of n and p materials. A few
electrons will be attracted across the junction, causing an electric field to be
established there. Current will then flow much more easily in one direction than
the other. When used in this way, the device is called a “rectifier”, useful for
changing alternating current (AC) to direct current (DC). In earlier days,
vacuum tubes filled this niche; but junction diodes are much smaller and
cheaper.

Transistors are 3 terminal devices, made of the same materials, in which the “gate”
terminal voltage or current controls flow between the other 2 terminals (“source”
and “drain”), leading to nearly everything in modern electronics and consumer
products. You may also see these referred to as “base”, “emitter”, and “collector”,
respectively. You are likely to own many little gismos containing millions of
transistors on a “chip”. They aren’t (yet) smarter than you; but a whole lot faster.
Electronic engineers today must study this quantum based theory ad nauseum.

The transistor was invented in 1947 – 1948 by three American physicists, John
Bardeen (1908 – 1991), Walter H. Brattain (1902 – 1987), and William B. Shockley
(1910 – 1989), at the American Telephone and Telegraph Company's Bell
Laboratories. In 1958 Jack St. Clair Kilby (1923 – 2005) of Texas Instruments, Inc.,
and Robert Norton Noyce (1927 – 1990) of Fairchild Semiconductor Corporation
independently thought of a way to reduce circuit size further. They laid very thin
paths of metal (usually aluminum or copper) directly on the same piece of material
as their devices. These small paths acted as wires. With this technique, an entire
circuit could be “integrated” on a single piece of solid material and an integrated
circuit (IC) thus created. ICs can contain hundreds of thousands or millions of
individual transistors on a single piece of material.
AT&T Bell Labs Physicists:
Bardeen, Brattain and Shockley
Replica of first Transistor

The phenomenon of quantum tunneling (see 19 Statistical Interpretation) has found
many practical applications; a few of which we’ll discuss here. These consist of a
pair of conductors, separated by an extremely thin insulator. Classically, negligible
current would flow between the conductors; but quantum mechanically, an
electron (or hole) reaching the insulator has some chance of appearing on the other
side. The resulting current depends very sensitively on the thickness of the
insulator, and on the voltage between the conductors.
Simplest, perhaps is the Esaki diode, over a range of voltages, the current falls as
the voltage is increased (negative resistance). As the insulator is extremely thin, the
device can amplify signals well into the microwave frequency region.
Negative differential resistance
Tunnel Diode Symbol

We’ll also mention the scanning tunneling microscope. Here, a very sharp needle is
scanned over a small region of a conducting surface. Teensy variations in height
cause big changes in the tunneling current, directly revealing atomic structure in the
scanned surface. The ability to see crystal structure and individual atoms has
advanced many areas of science.

Unlike semiconductors, the phenomenon of superconductivity preceded quantum
mechanics. Superconductivity was discovered in 1911 by the Dutch physicist Heike
Kamerlingh Onnes (1853 – 1926). Onnes found that the electrical resistivity of a
mercury wire disappears suddenly when it is cooled below a temperature of about
4 K (−269 °C or -459 °F). He won the Nobel Prize in 1913. The term “disappears” is
no exaggeration. If you drop a bar magnet onto a conducting surface, the changing
magnetic field causes a ring current (also called eddy current) in the conductor,
which causes a magnetic field that repels the magnet, noticeably reducing the fall
rate. An experiment in which a bar magnet was dropped into a superconducting
dish caused the magnet to float above the dish for several years, without dropping
perceptibly; so the resistance is literally zero. The turned up edge of the dish kept
the magnet from drifting away.
A magnet levitating above a high-
temperature superconductor,
cooled with liquid nitrogen.

Several other materials were found to become superconducting at different
temperatures, though none above about 23 K. Oddly, silver and copper, the best
conductors at room temperature, have no such transition – the resistance increases
without bound as the temperature drops. Another strange effect in
superconductors is that magnetic fields are completely expelled; though above a
certain critical field, it penetrates, and normal conductivity is restored.
Superconductivity was clearly well outside of classical physics; and so, should be
some kind of quantum effect. However, it wasn’t until 1957 that John Bardeen
(1908 – 1991) (again), Leon N. Cooper (1930 -), and John Robert Schrieffer (1931 -)
proposed what is now known as the BCS theory, which jointly won the 1972 Nobel
Prize. In it, “Cooper Pairs” of electrons with opposite spin, join in lockstep, and can
move together through the crystal lattice, without interacting, and thus
encounter no resistance.

Certain intermetallic compounds, such as Nb3Sn (3 niobium and 1 tin atom make up
the molecule), were found to have relatively high transition temperatures and quite
high tolerance for magnetic fields. They are useful for lossless power transmission
and lossless high field magnets. These properties are due to quite high field
penetration, not seen in pure metals.
In 1986, Karl Alexander Müller (1927 -) and Johannes Georg Bednorz (1950 -) found
the first of a group of complex metallic oxides, with the properties of ceramics, with
much higher transition temperatures. The current record is 134 K for
Hg2Ba2Ca2Cu3O8. Many laboratories are actively investigating these compounds and
searching for applications; but, as yet, nothing like BCS theory exists to explain their
behavior. Their work was good for the Nobel Prize in 1987.

One curious superconductive effect was predicted by Brian David Josephson in
1962. He said that, if a thin, weakly resistive, normal conductor were placed
between 2 superconductors, Cooper pairs would tunnel through; and a high
frequency oscillation would develop, depending on the applied voltage. This
property has since been observed in a wide variety of experiments; and the
arrangement is now known as a “Josephson Junction”. In an application called a
Squid (superconducting quantum interference device), we have the most sensitive
magnetic field detectors known. Josephson shared the 1973 Nobel Prize for Physics
with Leo Esaki and Ivar Giaever.
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Astrophysics is the study of the universe by observation and the application of
physical theories. The reverse is also true: physics is often learned by studying the
universe. Indeed, physics began with Johannes Kepler and Isaac Newton, based on
observations of the planets by Tycho Brahe (see 2 The Laws of Physics). Some of
what we know of the universe comes from looking and photographing through
telescopes; but the bulk of understanding has resulted from spectroscopy (see 9
Spectroscopy). By studying the spectra of stars, we have learned a great deal.
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24: Astrophysics

First, by identifying the various spectral lines we can tell what the star is made of –
at least the surface atoms that are radiating light and other electromagnetic
radiation. It turns out that there is an enormous range of stellar compositions,
which has led to an understanding of how stars evolve through nuclear burning
(fusion) of the light elements (see below, and also 26 Nuclear Physics). By and
large, stars are identified by spectral class: O, B, A, F, G, K, M. More or less, this is a
trend toward older, cooler, and redder. Every astronomy and astrophysics student
learns the mnemonic “Oh, be a fine girl, kiss me”. There’s a whole lot more to this;
but digression would become boredom.
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24: Astrophysics

Often, all the lines of a star’s spectrum are shifted a bit towards the violet,
compared to laboratory spectra of the same elements. This is caused by a Doppler
shift due to motion of the star in our direction, along the line of sight. A train
whistle dropping in frequency as it passes us is a similar effect (see our Doppler
Presentation occasionally shown on our Galaxy Stage). Similarly, if the lines are
shifted toward the red, the star is receding. The structure and mass distribution of
our galaxy, the Milky Way, has been determined almost entirely through this
technique. Other galaxies can also be studied, even when they are too remote to
be resolved into stars – the predominantly red shifts indicate that the universe is
expanding.
24: Astrophysics

Double and multiple stars are those which are gravitationally bound together, much
like the sun and planets of our solar system. When they are spread out enough,
they may be found from telescopic observations over several years. However, most
of these are too close to be resolved telescopically. Fortunately, the closer they
are, the faster they are moving about each other, and the greater is the difference
in Doppler shift. In this case, all the spectral lines are split up into one line for each
star; and we can learn a great deal about the system’s dynamics and stellar
composition.
24: Astrophysics

Sometimes, the lines are split, but differently for different lines. Clearly not a
Doppler shift, the splitting was eventually identified as the Zeeman effect, after
Pieter Zeeman (1865 – 1943), who observed it in laboratory spectra in 1896.
Zeeman shared the Nobel prize for this with Hendrick Antoon Lorentz (1853 – 1928)
in 1902. A real understanding of the Zeeman effect had to wait until quantum
mechanics showed that a magnetic field would sort atoms by the finite set of
energy states distinguished by the “m” quantum number (see 17 Periodic Table). In
short, this splitting told us about the strength of magnetic fields on the surfaces of
stars light years away. Certain “magnetic stars”, with large Zeeman splittings,
identified a whole new class of stars, and deepened our understanding of stellar
structure.
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24: Astrophysics
Zeeman Effect

What makes stars shine? The idea that stars are formed by the condensation of
gaseous clouds was part of the 19th century nebular hypothesis. The gravitational
energy released by this condensation could be transformed into heat, but
calculations by Hermann von Helmholtz (1821 – 1894) and Lord Kelvin (William
Thomson, Baron Kelvin of Largs, 1824 – 1907) indicated that this process could
provide energy to keep the sun shining for only about 20 million years. As the earth
was demonstrably far older, something else must be the energy source.
24: Astrophysics

In 1938, Hans Albrecht Bethe (1906 – 2005) (pronounced beta), proposed a
sequence of proton fusion reactions (see 26 Nuclear Physics) called “the carbon
cycle” by physicists. In it, a proton is absorbed by a carbon nucleus, becoming a
nitrogen nucleus; after absorbing 3 more protons, an excited oxygen nucleus is
formed, which immediately fissions into a helium nucleus (alpha particle) plus a
carbon nucleus, which can then absorb more protons. For the technically astute,
here is the full sequence:
12
C6 + 1
H1
13
N7 + γ; 13
N7
13
C6 + e+
; 13
C6 + 1
H1
14
N7 + γ;
14
N7 + 1
H1
15
O8 + γ; 15
O8
15
N7 + e+
; 15
N7 + 1
H1
16
O*
8
12
C6 + 4
He2
Together, we could compress the whole sequence as:
41
H1
4
He2 + 2e+
+ 3γ
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24: Astrophysics

Here, the subscripts are atomic numbers, i.e., the number of protons in a nucleus;
superscripts are rough atomic weights, essentially, the combined number of
protons plus neutrons in a nucleus; e+ is a positron or positive electron; and γ is a
gamma ray. The final O* indicates that the oxygen nucleus is created in an excited
state, and picks this way to unload its extra energy. Of course, Bethe knew that
each reaction had been observed experimentally. Go on, check the whole
sequence out to show that everything balances – it’s good for the soul.
24: Astrophysics
12
C6 + 1
H1
13
N7 + γ; 13
N7
13
C6 + e+
; 13
C6 + 1
H1
14
N7 + γ;
14
N7 + 1
H1
15
O8 + γ; 15
O8
15
N7 + e+
; 15
N7 + 1
H1
16
O*
8
12
C6 + 4
He2
41
H1
4
He2 + 2e+
+ 3γ

The set of reactions generates energy in 3 ways. First, the 3 gamma rays will
encounter electrons, causing Compton scattering, and eventually degrading to heat.
Second, the 2 positrons will encounter electrons and annihilate into more gamma
rays, yielding 511 kev each (see 26 Nuclear Physics). Finally, 4 protons together are
heavier than 1 alpha particle; the difference, by Einstein’s E = mc2
(see 20 Special
Relativity), appears as kinetic energy of the reacting particles.
Later, it was shown that this sequence occurs mostly in stars much hotter than the
sun and that the most important sequence there is:
1
H1 + 1
H1
2
H1 + e+
+ ν; 2
H1 + 1
H1
3
He2 + ν; 23
He2
4
He2 + 21
H1
and ν is a neutrino. There are alternate paths for the final reaction, that occur
mostly in hotter stars.
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When the hydrogen is largely exhausted, the star will shrink, and the center grows
hotter, till helium burning sets in. As these reactions release much less energy, the
helium exhausts fairly quickly, when several stages of further burning cause heavier
nuclei to form. Finally, iron is reached, when further fusion would consume energy,
rather than releasing it. The star then slowly cools and shrinks further, till the
“white dwarf” state is reached. Here, the pressures get so high that the electronic
shells of atoms are compressed into quantum degenerate states, like nothing on
earth. The quantum (and relativistic) theory of the structures of such stars was first
worked out by Subramanyan Chandrasekhar (1910 – 1995), who also predicted that
quantum degeneracy has an upper pressure limit, so that stars heavier than 1.44
suns would collapse further (into neutron stars or black holes). No white dwarfs
exceeding this limit have been found.
24: Astrophysics

Common objects found in our galaxy are “molecular clouds”, the most prominent
example being the “Great Nebula in Orion”, an easy binocular object in the Sword
of Orion. These are the birthplaces of stars; and spectroscopy has revealed a huge
range of chemical species in the gas and dust there; some, perhaps, involved in the
genesis of life. For some final comments on unsolved issues – dark matter and dark
energy – see 28 Theory of Everything.
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This is the page where we add to Heisenberg’s uncertainty, “strange” gets stretched
to “bizarre”; and physicists continue to argue. First, we’ll make a return trip into
“spin” (see 17 Matrix & Wave Mechanics). When 2 nuclear particles are banged
together in a particle accelerator, they often produce pi mesons or pions. These are
unstable; so, when they come to rest, they decay. Those with no charge decay into
an electron and a positron (an electron with positive charge), which fly apart in
opposite directions. This pion has spin 0; and as spin is rigorously conserved in
nuclear reactions, if the electron has spin ½, the positron spin must be -½, and vice
versa.
The stage is set for weirdness.
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25: Quantum Weirdness

By and by, after the split, we measure the spin of the particle on, say, the left. This
forces that particle’s wave function to collapse, and choose a spin direction. Then,
if the right particle’s spin is measured, it’s invariably found to be opposite, no
matter how far away it is. How did it “know” what its mate chose? Einstein called
this “spooky action at a distance”. In 1935, with 2 of his students, Boris Podolsky
and Nathan Rosen, argued that, if a measurement of one particle told you
something about another, a form of determinism must be involved, and thus
quantum mechanics must be incomplete. For them, this implied that when the
pion decayed, it gave its offspring some additional property, called a “hidden
variable”, which later influenced the spin choices. Many attempts were made to
formulate an enlarged quantum theory that would include such hidden variables,
without success; and many experiments were undertaken to shed further light.

Alas, in 1964, John Stewart Bell (1928 - 1990), an Irish theoretical physicist, proved
that no hidden variable theory could be consistent within quantum mechanics, a
result confirmed by several quite difficult experiments in the 1970’s. Does this
mean that if the 2 particles were a galaxy apart, you could measure one’s spin, and
force the other to be opposite? Apparently so. Does this violate special relativity
(see 20 Special Relativity)?
Apparently not; nothing physical goes from one to the other when the
measurements are made; and there is no way to use these measurements to
transfer information at greater than light speed. (We told the first particle to
choose its spin; but we didn’t tell it what to choose.) Bell’s result has since been
verified with photons and other particles.
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Einstein died in 1955; so we’ll never know what he would have thought of all this.
Maybe, when we revisit this program in the 22nd century, we’ll understand what
Bell’s theorem is really telling us. Then again, it might just get weirder.
This is a good place to throw in some numerology. The “fine structure constant” α
is a dimensionless product of some physical constants such as h and e. The
numerical values of most such constants depend on the units you choose; but for α,
the dimensions cancel out, and it’s a pure number. Early on, the value of 1/α ran
around 136; and much mumbo-jumbo was advanced to “explain” why it must be
just this. Sir Arthur Eddington even produced a “logical” derivation. Later, better
experiments gave a value closer to 137; and Eddington came up with a new
derivation. Alas, still more modern work yielded 1/α = 137.036. Fame awaits, if
you figure out why.

Let’s get back to uncertainty. In 17 Matrix & Wave Mechanics, we mentioned
Heisenberg’s Uncertainty Principle. This stated that, if you measure both a
particle’s position and momentum, then the product of their uncertainties can’t be
less than h/(4π). This may have seemed like an ad hoc assumption, added later to
the theory; but it can be rigorously proved from either matrix mechanics or
Schrödinger’s equation, plus the Born interpretation (see 19 Statistical
Interpretation).
So what does it mean? First, we aren’t talking about single particles; so we’ll need a
diversion into statistics. Suppose you measure all the weights of some large cohort
of frogs. Most likely, you’ll find a bell shaped distribution of results, with a few
teenys, a few bigs, and a bunch in the middle. Statisticians describe this spread by
the “standard deviation”, σ. For example, in the bell curve (or normal or
Gaussian distribution), a distance of σ on both sides of the mean would
include about 68% of all the frogs in the cohort.
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Now suppose we have a stream of particles passing by a device that measures their
position and momentum; then, the least possible value of the product of their
standard deviations, σposσmom is h/(4π), no matter how precise the measurements,
or the stream source. In other words, this minimum product of uncertainties is
intrinsic; although it’s important only on the atomic scale. Note that quantum
mechanics doesn’t put any limit on the possible accuracy of a single measurement,
say position. But, if you do a very precise position measurement, then the
measured momenta will have a large spread, and vice versa.

The uncertainty principle has a much wider application. We’ll cite only the energy –
time relation. A good example is the decay of radioactive nuclei. A well known
property of such nuclei is that half of any sample will decay in a time T called the
“half life”. If you then wait for an additional time T, half of the remaining sample
will decay, etc. What usually happens is that the decaying nucleus emits some sort
of particle, typically an electron or gamma ray. If one measures the energy of these
particles, one finds a spread. Since T is a measure of the uncertainty in the life of
the original nucleus, the product T σenergy generally comes out on the order of h; i.e.,
the shorter the half life, the wider the energy spread.

So why does the nucleus decay in the first place? Recall from 19 Statistical
Interpretation that the likelihood of finding the particle outside the nucleus
(tunneling) comes directly from the external wave function. Like the example given
there of a ball rolling inside a volcanic crater, if the nucleus could lose energy by
emitting a particle, then the wave function will have a non-zero value outside the
nucleus. The energy of the emitted particle depends on how far from the nucleus it
reappears, and thus on how much it’s affected by mutual repulsion.
Working out the theoretical energy spread for different radioactive nuclei,
comparing to measurements, and relating this to the measured half lives, probably
won someone a Ph. D., or at least a paper in a scholarly journal. Anyway, we’ll bet
it’s still weird.
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Radioactivity was first noticed by Antoine Henri Becquerel (1852 - 1908). By placing
uranium compounds on a photographic plate in a dark area in 1896, Becquerel
found that the plate had become blackened. This led to the well known story of the
Curies’ [Marie (1867 – 1934) and Pierre (1859 – 1906)] discovery of polonium and
radium 2 years later, and a series of other elements that uranium decayed into.
Since 1913, when Ernest Rutherford discovered that most of an atom’s mass is
concentrated in a tiny nucleus (see 14 Discovery of the Nucleus), attempts have
been made to apply quantum ideas to understand its structure. Much had to be
learned first. In 1919, Rutherford used alpha particles from radioactive decays to
bombard nitrogen. These produced other particles he soon identified as hydrogen
nuclei, and which he named “protons”. It soon became clear that each atomic
nucleus contained protons. Early models had protons making up essentially the
entire mass of the atom; and if there were Z electrons orbiting the nucleus, the
atomic number, then there must be additional electrons in the nucleus to
balance the charge, and make the atom electrically neutral.
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26: Nuclear Physics

This picture changed in 1932, when Sir James Chadwick (1891 – 1974), working in
Rutherford’s laboratory at Cambridge, used alpha particles to bombard beryllium.
What emerged was a neutral particle, of about the same mass as the proton, and
now called a “neutron”. In this picture, there were just Z protons; and the
remaining mass is made up by neutrons.
1932 also saw another landmark in nuclear physics – the development of the
Cockroft – Walton generator, the first particle accelerator. The inventors were Sir
John Douglas Cockroft (1897 – 1967) and Ernest Thomas Sinton Walton (1903 –
1995). For the electrically astute, their device worked by charging, say N capacitors
in parallel; then, with a bunch of switches, reconnecting them in series, thus
multiplying the voltage by N. Don’t try this at home. The first nuclear reactions
produced without radioactives were the result.
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26: Nuclear Physics

The 1930’s were a truly fertile time for nuclear physics. Robert Jemison Van de
Graaff (1901 – 1967) slowly improved another kind of generator, based on
transferring charge from a low to a high potential by means of a moving, circulating,
insulated belt. These machines are limited to millions of volts by electrical
breakdown; but they have the advantage of operating continuously. Since then,
they have seen a lot of use as injectors in higher energy devices. The 1930’s
also saw the development of the cyclotron (see 26 Cyclotrons).
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26: Nuclear Physics
Cockcroft-Walton voltage multiplier
Van de Graaff generator

The energies reached by these devices are generally measured in “electron volts” –
1 ev is the energy change seen by an electron or proton passing between 2 plates
with 1 volt applied between them. The Cockroft – Walton generator typically
operated at a few hundred kev (k = kilo = 1000); and the Van de Graaff generators
and early cyclotrons (see 26 Cyclotrons) at a few Mev (M = Mega = million). By the
1950’s accelerators had reached the Gev range (G = Giga = billion); and in the
1970’s, into the Tev (T = Tera = trillion). Colliding 2 beams traveling in opposite
directions is far more effective at exciting truly high energy events; so the Large
Hadron Collider at CERN on the French – Swiss border is expected to shortly reach 7
Tev.
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26: Nuclear Physics
Large Hadron Collider

So what does all this mean for physics? The first question faced was how could a
nucleus, composed largely of positively charged particles, not just fly apart through
electrostatic repulsion? At the time, there were only 2 known forces between
particles – electromagnetic and gravitational – the former clearly ruled out, and the
latter far too weak. Evidently, a “strong force” must exist between nucleons.
In 1935, Hideki Yukawa1
(1907 – 1981) suggested that the strong force was the
result of new particles being exchanged between nucleons. He predicted their
mass, and that if free, would decay with a particular very short half life. The next
year, Carl David Anderson (1905 – 1991) discovered a particle in cosmic rays which
he thought to be Yukawa’s particle. Alas, it was a bit short of mass, had far too long
a half life, and penetrated deep into the earth, showing that it interacted much
too weakly with nuclei. Anyway, we now call it a mu meson, or muon; and we
will encounter it again in 27 Quantum Chromodynamics.
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26: Nuclear Physics
1
In Japan, this would be Yukawa Hideki, in line with their practice of putting the family name first.

In 1947, Cecil Frank Powell (1903 – 1969) found Yukawa’s particle in high altitude
cosmic ray emulsions. Called the pi meson, or pion, it closely matched Yukawa’s
predictions, and decayed into Anderson’s muon. The bearer of the strong force had
been found, or so it was thought (see 27 Quantum Chromodynamics).
Following the discovery of the neutron, it was natural to ask whether nuclear
structure could be worked out along the lines of the quantum mechanical picture of
electron energy levels. Such a description was developed independently in the late
1940’s by the American physicist Maria Goeppert Mayer (1906 – 1972), and the
German physicist Johannes Hans Daniel Jensen (1907 – 1973), who shared the
Nobel Prize for Physics in 1963 for their work, along with Eugene P. Wigner.
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26: Nuclear Physics

In the shell nuclear model, the constituent nuclear particles are paired neutron with
neutron, and proton with proton, in nuclear-energy levels that are filled, or closed,
when the number of protons or neutrons equals 2, 8, 20, 28, 50, 82, or 126, the so-
called magic numbers that indicate especially stable nuclei. That these are all even
numbers is because both protons and neutrons have spin ½. The unpaired
neutrons and protons account for the properties of a particular species of nucleus
as valence electrons account for the chemical properties of the various elements.
The shell model accurately predicts certain properties of normal nuclei, such as
their angular momentum; but for nuclei in highly unstable states, the shell model is
no longer adequate, and must be modified or replaced by another model, such as
the liquid-drop model, collective model, compound-nucleus model, or optical
model. Suffice it to say that a really solid model of the nucleus is yet to come.
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26: Nuclear Physics

If you bang one nuclear particle off another in an accelerator, and see stuff
emerging, you can generally measure the energy and momenta of all the particles.
Studies of radioactive decays of many unstable nuclei often produced beta rays
(electrons). By observing both the electron and the recoiling nucleus, it was found
that the electrons had a spread of energies, and the combined momentum and
spin weren’t conserved.
Since charge was conserved, Wolfgang Pauli (1900 – 1958) (see 17 Pauli) proposed
in 1931 that the missing energy and momentum is carried away from the nucleus by
some particle (later named the neutrino) that’s uncharged, has little or no mass, has
spin ½, and had gone unnoticed because it interacts with matter so seldom that it’s
nearly impossible to detect. This particle wasn’t observed until 1956, when
Clyde L. Cowan Jr. (1919–1974) and Frederick Reines (1918 – 1998) looked for it
close to a large nuclear reactor, correlating their results with reactor activity.
Hard as they are to see (most zip through the whole earth), they reinstated the
conservation laws, and opened a new chapter in physics.
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26: Nuclear Physics

Nuclear physics is an endless subject; but we’ll conclude with just 2 more topics,
nuclear fusion and fission, both processes that can release large amounts of energy.
The fusion of 2 light nuclei to produce another, with around 27 protons (iron) or
less will give off energy, and is what powers stars (see 24 Astrophysics, where
several such reactions are listed). This is the basis for H-bombs, and a possible
source of electrical power (see 26 Nuclear Applications).
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26: Nuclear Physics

On the other hand, fission depends on breaking up heavy nuclei into pieces, again
somewhere around 27 protons – the most stable nuclei. Following the discovery of
the neutron by James Chadwick (see above), Enrico Fermi (1901 – 1954) and his
associates tried bombarding a variety of nuclei with them a few years later. While
originally believing some transuranic elements had been produced, Otto Hahn
(1879 – 1968) and Fritz Strassmann (1902 – 1980) found fission fragments in the
debris. This work was carefully nailed down by Lise Meitner (1978 – 1968) and Otto
Robert Frisch (1904 – 1979), just before World War II, and the “Nuclear Era” had
begun (see 26 Nuclear Applications).
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26: Nuclear Physics
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Nuclear physics has produced a lot more than bombs; but let’s get those out of the
way first. Following Meitner and Frisch, the Manhattan Project worked on the idea
that, when a neutron caused a nucleus to fission (see 26 Nuclear Physics), from 1 to
3 neutrons were released, along with the fission products. In a relatively pure
fissionable material – 235
U92, and some isotopes of plutonium or thorium, these
could cause new fissions, leading to a “chain reaction”, and a huge release of
energy – the “atomic bomb”. If you’re interested in how to blow up your school (or
your city), our advice is to look elsewhere.
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26: Nuclear Applications
Uranium-235 Fission Chain Reaction

“Hydrogen bombs” are different. In stars, fusion reactions require high density, and
very hot reactants. The high temperature (millions of degrees) is so that the
electrons and nuclei roam freely in a “plasma”, and that a few nuclear velocities are
high enough to penetrate another’s electrostatic repulsion, and get close enough to
fuse. In H-bombs, tritium (an unstable isotope of hydrogen with 2 neutrons in the
nucleus) is packed around a fission bomb “trigger”. When the trigger goes off, the
temperature of the tritium is instantly raised into the hundreds of millions of
degrees, much hotter even than stellar centers, when the tritium will fuse to
generate still more energy. In fact, with enough tritium, there’s no limit to the
energy release. It’s fortunate that, with a half life of only 12 years, there’s no
tritium lying around for us to make trouble with.

Probably the most important application of fission technology today is the
generation of electrical power.
The first nuclear reactor was built of slugs of uranium, surrounded by uranium
oxide, and embedded in bricks of pure graphite (the carbon that is used in “lead”
pencils). The point of the graphite is to slow down (moderate) the fast neutrons, so
that they are more readily absorbed in uranium. In the language of physics, the
fission cross section of an isotope of uranium – 235
U92 – increases as you lower the
neutron kinetic energy. At the same time, the cross section for absorption in the
most common isotope - 238
U92 – also increases. The resulting 239
U92 quickly decays by
beta emission to neptunium – 239
Np93 – and from there to plutonium – 239
Pu94 – a
readily fissionable substance.

The first reactor, under the aegis of Enrico Fermi (1901 – 1954), achieved a self
sustaining reaction in late 1942. “The Italian Navigator has reached the New
World.”
Today, a significant fraction of the world’s electrical power is generated by nuclear
reactors. While producing no CO2, or other contributors to global warming, reactors
do generate radioactive fission products. Probably the biggest impediment to
expanding this energy source is what to do with them – all present options are
expensive, and potentially hazardous. Unfortunately, all our past administrations
and Congresses have been unwavering in their refusal to deal effectively with the
issue, so the stuff piles up at reactor sites.
1951: Experimental Breeder Reactor-I lit these four
lightbulbs to demonstrate the first usable amount of
electricity from nuclear energy.

The other possibility for nuclear power is controlled fusion, for which there are 2
roads presently being pursued – magnetic confinement and inertial confinement.
In either case, the object is to get P, the product of temperature, density, and
confinement time high enough for significant fusion to take place. In magnetic
confinement, some arrangement of magnets, and particle or plasma injectors, must
not only create a high enough P, it must keep the hot plasma away from the walls.
Failure to do this wouldn’t just damp the reaction, it would slag down the reactor.
Fusion is generally accompanied by the emission of 14 Mev neutrons, the number
of which tells you how well you are doing.
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Over the last few decades, fusion reactors have become increasingly sophisticated,
and have reached ever higher P. A next major step in the development of fusion
power is the construction of a facility to study the physics of a burning, ignited
plasma. It is anticipated that this would occur in a planned new experiment, the
International Thermonuclear Experimental Reactor (ITER). This is a very large
experiment that would investigate both the physics of an ignited plasma and
reactor technology.
ITER would generate about 1.5 billion watts of thermal fusion power (which would
not be converted to electricity). The cost of the device (in the range of $10 billion)
has encouraged international collaboration; and from its conception ITER has
involved as equal partners the European Union, Japan, Russia, and the United
States. Operation is to begin about 2010. It is hoped that ITER would be
followed by a demonstration fusion reactor power plant. Besides generating
no greenhouse gasses, fusion reactors should produce only a tiny fraction of
the radioactive waste as fission reactors.
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In inertial confinement, a tiny blob of solid mixed deuterium and tritium is suddenly
hit by a large number of simultaneous intense laser pulses, causing a compression
wave to head toward the center of the blob, producing a large P before the blob
flies apart. Again, there has been real progress toward higher P. The latest is called
the National Ignition Facility, underway at the Lawrence Livermore Laboratory, and
expected to reach ignition. Unfortunately, this facility is funded primarily for
weapons research.
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Deuterium-Tritium Fusion

Besides the above, there’s a host of applications of nuclear technology; we’ll
mention a few, in no particular order. One of these is dating events in remote and
recent past.
The radioactive decay sequence starting with 238
U92 begins with an alpha decay with
a half life of about 4 billion years. There follows several more steps, with a stop at
radium, and ending at lead, specifically, the stable isotope 206
Pb82. Ancient rocks can
be dated from the ratio of the amount of this isotope to that of 238
U92, after
correcting for any preexisting lead. A problem with this is that one intermediate
product, radium, decays into radon, an inert gas at room temperature. Although
the half life of this radon in only 4 days, to the extent that it escapes, the
206
Pb82/238
U92 ratio is thrown off. There are many other “isotope clocks”, useful for
more recent rock formations, and for dating meteorites from the formation of
the solar system, and for pieces of Mars and the Moon, blasted into space at
some earlier time.
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In this vein, we must mention “carbon dating”. Cosmic rays impinging on the upper
atmosphere generate a small amount of 14
C6, unstable with a half life of about
14,000 years. Some of this is incorporated into growing plants, leading to a tiny,
but fairly stable proportion of the total carbon. When the plant dies, there is no
further replenishment, and the proportion decays exponentially. By measuring this
fraction in buried trees, or papyrus, or charcoal, we can determine when the plant
died. The value of this, and related techniques, to archeology can’t be overstated.
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Some isotopes such as 60
Co27 and 90
Sr38 are strong gamma emitters, and are useful in
penetrating heavy steel plates and the like, and also for sterilizing foods. Biological
research is often aided by introducing a rare isotope into a “tracer” compound, and
seeing where it winds up in a plant or animal, either by the radiation it emits, or by
mass spectroscopy if the isotope is stable. A few isotopes are positron emitters,
useful because the positrons annihilate with electrons, and produce readily
identifiable 511 kev gamma rays. “PET scans” (positron emission tomography) rely
on this technique to identify a whole range of biological activities in the body.
Finally, cancer treatments often involve inserting tiny capsules of isotopes with
short range radiation emitters into affected organs; or, for harder to reach organs,
tracer compounds can carry unstable isotopes to affected places.
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Cyclotrons are devices in which charged particles (usually protons) are caused to
follow a circular path by an external magnetic field.
The first were developed by Ernest Orlando Lawrence (1901 – 1958) and his student
M. Stanley Livingston at the University of California, Berkeley, in the early 1930’s.
Their 30 cm machine generated energies of about 1 Mev, and established that it
was possible to achieve the energies needed to penetrate the nucleus of an atom.
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26: Cyclotrons
Diagram of cyclotron operation
from Lawrence’s 1934 patent

At low energies, the “cyclotron frequency” – the frequency that a charged particle
will orbit in a uniform magnetic field - is independent of the energy – only the path
radius increases with increasing energy. This path is mostly inside of a pair of
hollow D shaped electrodes facing each other (see figure). An oscillating voltage is
applied between the D’s at twice the cyclotron frequency. The particles drift inside
each D, but, if they arrive at the gap at the right time, they are accelerated each
time they cross one of the gaps. With gradually increasing energy, they spiral
outward, until extracted near the rim, and are allowed to strike a target. Lawrence
received the 1939 Nobel Prize.
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26: Cyclotrons

The cyclotron soon became the leading instrument used to study nuclear physics. It
also became a primary source for the creation of radioactive materials used in the
medical field as diagnostic tracers, and for cancer treatments. The cyclotron led to
the discovery of neptunium and plutonium, the first artificial elements ever
produced.
A limitation on the early designs was caused by the relativistic increase in particle
mass with increasing energy. As discussed in 20 Special Relativity, the mass of a
particle increases above its rest mass as the velocity approaches the speed of light
thus decreasing the cyclotron frequency. Since the rest mass of a proton is about 1
Gev, the practical limit of the early machines was about 25 Mev.
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26: Cyclotrons

A solution to this problem was to gradually increase the confining magnetic field as
the particle mass increases, thus holding the cyclotron frequency constant. Such
devices are called “synchrotrons”. They were first proposed by Vladimir Iosifovich
Veksler (1907 – 1966) in 1944, and Edwin Mattison McMillan (1907 – 1991) in 1945.
McMillan and Glenn Theodore Seaborg (1912 - 1999) were the discoverers of
neptunium and plutonium in 1940, and were together awarded the 1951 Nobel
Prize in Chemistry.
With the addition of accelerating field frequency modulation and various
improvements in the confining magnetic field design, larger and larger machines
reached higher energies, today into the Tev range (T = trillion).
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26: Cyclotrons

In these latter machines, a new problem arises. A particle with a Tev of energy is
much more massive than a stationary nuclear target (1 Gev per nucleon); so the
debris from the collision heads down track, causing most of the original energy to
be wasted as kinetic energy of motion, rather than generating new particles.
A solution to this is to generate 2 beams circling in opposite directions in almost the
same track. The beams are caused to intersect at a few locations around the circle,
where large, complicated, and expensive detectors are located to analyze the
debris. The latest such design is the Large Hadron Collider at CERN, on the French-
Swiss border, now in the testing phase, with energies up to 7 Tev. The beam tube is
17 kilometers in diameter.
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26: Cyclotrons

Much of the reason for the large size is that, to make particles move in other than a
straight line, you have to apply a force to them (here the magnetic field), and thus
continually accelerate them. Since Maxwell’s time (see 7 Electromagnetic Theory),
we have known that accelerated electric charges will radiate, and dissipate some of
the energy we’ve gone to such pains to acquire. The bigger the circle, the less
radiation loss; so, if you’re a really big thinker, imagine what we’ll be able to do if
the beam tube could encircle the earth, or maybe the earth’s orbit around the sun.
For electrons, the radiation problem is much worse than for protons or heavier
particles, causing physicists to go to linear accelerators, where the electrons go
through a straight beam tube only once. We could elaborate on this, but this green
box is about cyclotrons; and we are wandering from quantum mechanics.
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In the years following the discovery of the pion, a bewildering host of new particles
were found, leading most physicists to believe that few of these could be
fundamental. In 1961, Murray Gell-Mann (1929 -) and Yuval Ne’eman (1925 –
2006) organized all the particles into sets containing 1, 8, 10, or 27 members,
reminiscent of the organization of electrons into shells (see 17 Periodic Table). As
most of these sets had 8 members, Gell-Mann called this system the “Eight Fold
Way”.
Central to this system was the idea that protons and neutrons are actually made up
of more basic particles Gell-Mann called “quarks”, a term he borrowed from James
Joyce’s “Finnegan’s Wake”. The simplest quarks are called “up” and “down” (u and
d). Letting e be the charge of the electron (negative), then u has a charge of
-2e/3, and d has e/3. Thus, a proton, with a charge of –e, is composed of uud,
and the uncharged neutron is udd. The corresponding antiparticles are made
of antiquarks ū and đ with opposite charges.
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27: Quantum Chromodynamics

The pions are all made from quark-antiquark pairs. The negative pion, π-
is really
ūd; the positive pion, π+
is uđ; and the neutral pion, π0
dithers between uū and dđ.
The pions are all unstable. π+ and π- decay into muons, µ+ and µ -, plus neutrinos;
which, in turn decay into positrons and electrons plus more neutrinos. π0
either
dissolves into 2 gamma rays, or an electron and a positron (this is really peculiar,
see 25 Quantum Weirdness).
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In Gell-Mann’s scheme, quarks actually come in 6 “flavors” (invoking such a term is,
well, a matter of taste): up, down, strange, charm, top, and bottom. The latter are
also called truth and beauty, more examples of physicist’s whimsy. Sets of these,
together with their corresponding antiquarks, may be combined into a battalion of
unstable particles, collectively called mesons, and organized as part of the Eight-
Fold Way. Enlarging and understanding this battalion by beating things together in
accelerators, and seeing what emerges, gives much amusement and remuneration
to particle physicists.
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Artist sketch of mesons inside nuclei

Before pressing further into this jungle, we should pause to examine those
creatures that contain no quarks. First of course is the electron. In no end of
accelerator runs, at a huge range of energies, with all sorts of targets, the electron
hasn’t shown any internal structure. So far as we know, its sole properties are its
charge e, and spin ½. It is the first of a class of particles we call “leptons” that aren’t
affected by the strong force that holds nuclei together (see 26 Nuclear Physics).
The first grouping of leptons consists of the electron, the electron neutrino νe
, and
their antiparticles. The second grouping has the mu meson, or muon µ, that we
encountered in pion decay in 26 Nuclear Physics. Along with this is the mu neutrino
νµ and the usual suspect antiparticles. Finally, there is the tau meson τ, the tau
neutrino ντ and their antiparticles. The mesons have charge ±e; and all leptons
have spin ±½. As for mass in electron volts, the electron has 511 kev, the muon
106 Mev, and the tau meson 1.86 Gev. The neutrinos are massless, or nearly so.
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Massless? A variety of direct measurements failed to detect any; but 1 pesky
observation lead to rethinking. Measurements of electron neutrinos coming from
the sun fell short of that expected from fusion reactions (see 24 Astrophysics) by
about a factor of 3. The most popular current hypothesis is that νe can change into
νµ or ντ in flight, when we could only detect a third of those coming by. If it were
really massless, it would travel at the speed of light; its clock would stop (see 20
Special Relativity); and it couldn’t change form. So, today, we think neutrinos have
some tiny rest mass; but we haven’t been able to measure it.
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Ground-based telescopes, like the Anglo-Australian
Observatory, saw the light from supernova 1987A
several hours after the Kamiokande and IMB
experiments had already detected the neutrinos that
were emitted.

In the Eight-Fold Way, quarks also possess a property called “color”. Of course, you
can’t see color any more than you can taste flavor. There are 3 colors – red, green,
and blue. Hadrons and mesons (those things made up of quarks) all have “white”
combinations – all 3 colors in protons and neutrons, and a color and its anticolor in
mesons. You may disagree; but physicists find great beauty in all this.
If pions are the mediator of the strong force that makes protons stick together in
nuclei, what holds protons together? They are, after all, made up of charged
quarks. Enter, stage right, the gluon. One reason for doing this is that accelerator
debris never contained a free quark; and oil drop like experiments (see 11 Discovery
of Electrons) never found a charge smaller than e. We say that quarks are held
together by exchanging gluons, which took over the strong force role from
pions (see 26 Nuclear Physics), just as atoms are held together by virtual
photons (see 21 Quantum Electrodynamics).
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In the latest view (or so we think), a gluon contains a color plus a different anticolor.
A pair of more complex arrangements make a total of 8 kinds of gluons. Gluons
can change a quark’s color; e.g., if a green-antired gluon meets a red quark, it’s
absorbed, leaving a green quark. Or a green-antired gluon can join with a red-
antiblue gluon, leaving a green-antiblue gluon. It’s this blizzard of gluons that bind
quarks together in protons and neutrons, and also holds these particles together in
nuclei.
So, what about the pions? Well, if you bash protons, etc. together in accelerators,
you would expect to find quarks among the debris; but most of what comes out is
pions. Raise the energy, and you get heavier and more energetic mesons, plus
standard nuclear fragments. Since the pions are quark-antiquark pairs, you
get quarks all right, but never alone.
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Just like in QED (see 21 Quantum Electrodynamics), we can compute particle
properties by considering all the possible pairs of interactions, occurring at all
points in spacetime; but, while you can correct this by adding all the 4 interaction
scenarios; unlike QED, the likelihood doesn’t fall off anywhere near as fast; and you
need to consider far more complex interactions to get similar accuracy in the
theoretical estimates of properties. Today, theory and experiment are in around
10% agreement, unsatisfactory at best. Stay tuned; but if all this makes you want to
go find a beer, you are far from the first.
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Quantum Chromodynamics, also known as the Eight-Fold Way, was jointly worked
out by Murray Gell-Mann (1929 -) and Yuval Ne’eman (1925 – 2006). Having
entered Yale University at the age of 15, Gell-Mann received his B.S. in physics in
1948 and his Ph.D. at the MIT in 1951. His doctoral research was on subatomic
particles. In 1952 Gell-Mann joined the Institute for Nuclear Studies at the
University of Chicago. The following year he introduced the concept of
“strangeness”, a quantum property that accounted for previously puzzling decay
patterns of certain mesons. As defined by Gell-Mann, strangeness is conserved
when any subatomic particle interacts via the strong force - i.e., the force that binds
the components of the atomic nucleus.
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27: Gell-mann and Ne’eman

In 1961 Gell-Mann and Ne'eman, an Israeli theoretical physicist, independently
proposed a scheme for classifying previously discovered strongly interacting
particles into a simple, orderly arrangement of families. Called the Eight-Fold Way
(after Buddha's Eightfold Path to Enlightenment and Bliss), the scheme grouped
mesons and baryons (e.g., protons and neutrons) into multiplets of 1, 8, 10, or 27
members on the basis of various properties. All particles in the same multiplet are
to be thought of as variant states of the same basic particle. Gell-Mann speculated
that it should be possible to explain certain properties of known particles in terms
of even more fundamental particles, or building blocks. He later called these basic
bits of matter “quarks”, adopting the fanciful term from James Joyce's novel
Finnegans Wake. One of the early successes of Gell-Mann's quark hypothesis was
the prediction and subsequent discovery of the omega-minus particle (1964).
Over the years, research has yielded other findings that have led to the wide
acceptance and elaboration of the quark concept.
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Gell-Mann joined the faculty of the California Institute of Technology, Pasadena in
1955, and was appointed Millikan professor of theoretical physics in 1967
(emeritus, 1993). He published a number of works, notable among which are The
Eightfold Way (1964), written in collaboration with Ne'eman; Broken Scale Variance
and the Light Cone (1971), coauthored with K. Wilson; and The Quark and the
Jaguar (1994).
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Ne'eman studied engineering at the Technion – Israel Institute of Technology, Haifa,
and Imperial College, London, and was an officer in the Israel Defense Force for 12
years. He was a member (1952 – 1961) of Israel's Nuclear Energy Commission and
was scientific director (1961 – 1963) of the Nahal Soreq nuclear reactor.
In 1965 he founded Tel Aviv University's School of Physics and Astronomy, which he
directed until 1972; he also served as the university's president (1971 – 1975).
Ne'eman was a founder (1979) of the right-wing Tehiya party and represented that
party in the Knesset (parliament) for three terms (1982 – 1992). He was the
founder of the Israel Space Agency in 1983. He served as Israel's first minister of
science and development (1982 – 1984) and held the post again in 1990 – 1992. He
was awarded the Israel Prize for his scientific work in 1969.
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In this final box, we’ll talk about the future – unsolved problems and the dreams of
physicists and astrophysicists. Following Newton, over the next 2 centuries, the
notion of the conservation of energy was gradually broadened to include
mechanical, gravitational, chemical, and electrical energies (see 5 Conservation of
Energy and 6 Thermodynamics & Statistical Mechanics). This amounted to a
consolidation of physical ideas. Around 1860, Maxwell published the Maxwell
equations, unifying light, electrostatics, and magnetic phenomena (see 7
Electromagnetic Theory).
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28: Theory of Everything

In 1905, special relativity unified space and time, and also mass and energy. The
latter meant that the conservation laws for mass and energy had become a single
law. For much of his life, Einstein looked for a way to unify electromagnetism and
gravity (unified field theory), without success. During the 1960’s and 1970’s several
physicists constructed an “electroweak” theory, unifying electromagnetism with the
“weak force” involved in radioactivity.
Throughout the 20th century, it became ever more clear that general relativity and
quantum mechanics are not compatible; and the notion that a single theory should
encompass all of the above acquired the moniker “A Theory of Everything”.
Currently (we think), “string theory” or “superstring theory” holds the spotlight.
These are a grab bag of mathematical constructs, accorded great beauty by their
proponents. The general ideas have everything made of itty bitty vibrating
strings, that live in 11 or 12 dimensions, most of which are wrapped up so we
don’t notice them. Alas, to date, none have produced predictions, testable
with today’s instruments.
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This is probably a good place (maybe the only place) to mention “dark matter” and
“dark energy”. The former surfaced when increasingly comprehensive spectral
measurements of stellar doppler shifts in our galaxy and elsewhere (see 24
Astrophysics) showed that the orbital speeds of stars around the galactic centers
were too high, indicating that there is more mass in the galaxy than we can see. Of
course, calling this “missing” or “dark” matter explains nothing; but there are lots of
theories as to what’s involved. One thing seems clear – the stuff, whatever it is,
appears to concentrate around galactic centers; i.e., there seems to be less of it in
intergalactic space.
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Dark energy is even more mysterious. Light from distant galaxies comes from the
past. A galaxy a billion light years away is seen as it was a billion years ago. This is
generally red shifted (doppler shift seen in galactic spectra – see 9 Spectroscopy
and 24 Astrophysics) by an amount that increases, more or less, with distance.
Since the late 1920’s we have known from this that the universe is expanding, ever
since the “big bang” about 13.6 billion years ago. More refined measurements,
together with general relativity have more recently shown that, contrary to what
we all thought, the expansion is accelerating.
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When the field equations of general relativity were solved for a finite universe, it
was shown that a static solution is unstable, implying that it must be either
expanding or contracting. As it was then regarded as static and unchanging,
Einstein added a “cosmological constant L” to the field equations. When expansion
was found, Einstein called  “the worst mistake of my life”, and disavowed it. Now,
with an accelerating expansion,  has been revived; but just because we have an
equation that allows acceleration, doesn’t mean we understand it. So, today, we
have a new mystery, labeled “Dark Energy”. Now, though we don’t understand; our
ignorance has a name.
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