Physics Module 2 .pdf on quantum physics

Need for Quantum Physics
•Problems remained from classical mechanics that
the special theory of relativity didn’t explain.
•Attempts to apply the laws of classical physics to
explain the behavior of matter on the atomic scale
were consistently unsuccessful.
•Problems included:
– Blackbody radiation
• The electromagnetic radiation emitted by a heated object
– Photoelectric effect
• Emission of electrons by an illuminated metal
Introduction

Quantum Mechanics Revolution
•Between 1900 and 1930, another revolution
took place in physics.
•A new theory called quantum mechanics was
successful in explaining the behavior of particles
of microscopic size.
•The first explanation using quantum theory was
introduced by Max Planck.
– Many other physicists were involved in other
subsequent developments
Introduction

Blackbody Radiation
•An object at any temperature is known to emit thermal
radiation.
– Characteristics depend on the temperature and surface
properties.
– The thermal radiation consists of a continuous distribution
of wavelengths from all portions of the em spectrum.
•At room temperature, the wavelengths of the thermal
radiation are mainly in the infrared region.
•As the surface temperature increases, the wavelength
changes.
– It will glow red and eventually white.
Section 40.1

Blackbody Radiation, cont.
•The basic problem was in understanding the
observed distribution in the radiation emitted by
a black body.
– Classical physics didn’t adequately describe the
observed distribution.
•A black body is an ideal system that absorbs all
radiation incident on it.
•The electromagnetic radiation emitted by a
black body is called blackbody radiation.
Section 40.1

Blackbody Approximation
•A good approximation of a
black body is a small hole
leading to the inside of a
hollow object.
•The hole acts as a perfect
absorber.
•The nature of the
radiation leaving the cavity
through the hole depends
only on the temperature of
the cavity.
Section 40.1

Blackbody Experiment Results
•The total power of the emitted radiation increases
with temperature.
– Stefan’s law
P = s A e T4
– The emissivity, e, of a black body is 1, exactly
•The peak of the wavelength distribution shifts to
shorter wavelengths as the temperature increases.
– Wien’s displacement law
– lmaxT = 2.898 x 10-3 m . K
Section 40.1

Intensity of Blackbody Radiation,
Summary
•The intensity increases
with increasing
temperature.
•The amount of
radiation emitted
increases with
increasing temperature.
– The area under the
curve
•The peak wavelength
decreases with
increasing temperature.
Section 40.1

Rayleigh-Jeans Law
•An early classical attempt to explain blackbody
radiation was the Rayleigh-Jeans law.
•At long wavelengths, the law matched
experimental results fairly well.
 
I ,  4
2 B
π c k T
λ T
λ
Section 40.1

Rayleigh-Jeans Law, cont.
•At short wavelengths,
there was a major
disagreement between the
Rayleigh-Jeans law and
experiment.
•This mismatch became
known as the ultraviolet
catastrophe.
– You would have infinite
energy as the
wavelength approaches
zero.
Section 40.1

Max Planck
•1858 – 1847
•German physicist
•Introduced the concept
of “quantum of action”
•In 1918 he was
awarded the Nobel Prize
for the discovery of the
quantized nature of
energy.
Section 40.1

Planck’s Theory of Blackbody Radiation
•In 1900 Planck developed a theory of
blackbody radiation that leads to an equation
for the intensity of the radiation.
•This equation is in complete agreement with
experimental observations.
•He assumed the cavity radiation came from
atomic oscillations in the cavity walls.
•Planck made two assumptions about the nature
of the oscillators in the cavity walls.
Section 40.1

Planck’s Assumption, 1
•The energy of an oscillator can have only certain
discrete values En.
– En = n h ƒ
• n is a positive integer called the quantum number
• ƒ is the frequency of oscillation
• h is Planck’s constant
– This says the energy is quantized.
– Each discrete energy value corresponds to a different
quantum state.
• Each quantum state is represented by the quantum number,
n.
Section 40.1

Planck’s Assumption, 2
•The oscillators emit or absorb energy when
making a transition from one quantum state to
another.
– The entire energy difference between the initial
and final states in the transition is emitted or
absorbed as a single quantum of radiation.
– An oscillator emits or absorbs energy only when it
changes quantum states.
– The energy carried by the quantum of radiation is
E = h ƒ.
Section 40.1

Energy-Level Diagram
•An energy-level diagram
shows the quantized
energy levels and allowed
transitions.
•Energy is on the vertical
axis.
•Horizontal lines represent
the allowed energy levels.
•The double-headed
arrows indicate allowed
transitions.
Section 40.1

More About Planck’s Model
•The average energy of a wave is the average
energy difference between levels of the
oscillator, weighted according to the probability
of the wave being emitted.
•This weighting is described by the Boltzmann
distribution law and gives the probability of a
state being occupied as being proportional to
where E is the energy of the state.
B
E k T
e
Section 40.1

Planck’s Model, Graph
Section 40.1

Planck’s Wavelength Distribution
Function
•Planck generated a theoretical expression for the
wavelength distribution.
– h = 6.626 x 10-34 J.s
– h is a fundamental constant of nature.
•At long wavelengths, Planck’s equation reduces to the
Rayleigh-Jeans expression.
•At short wavelengths, it predicts an exponential
decrease in intensity with decreasing wavelength.
– This is in agreement with experimental results.
 
 
I , 

2
5
2
1
B
hc λk T
πhc
λ T
λ e
Section 40.1

Einstein and Planck’s Results
•Einstein rederived Planck’s results by assuming
the oscillations of the electromagnetic field
were themselves quantized.
•In other words, Einstein proposed that
quantization is a fundamental property of light
and other electromagnetic radiation.
•This led to the concept of photons.
Section 40.1

Photoelectric Effect
•The photoelectric effect occurs when light
incident on certain metallic surfaces causes
electrons to be emitted from those surfaces.
– The emitted electrons are called photoelectrons.
• They are no different than other electrons.
• The name is given because of their ejection from a
metal by light in the photoelectric effect
Section 40.2

Photoelectric Effect Apparatus
•When the tube is kept in
the dark, the ammeter
reads zero.
•When plate E is
illuminated by light having
an appropriate wavelength,
a current is detected by the
ammeter.
•The current arises from
photoelectrons emitted
from the negative plate and
collected at the positive
plate.
Section 40.2

Photoelectric Effect, Results
•At large values of DV, the
current reaches a
maximum value.
– All the electrons emitted
at E are collected at C.
•The maximum current
increases as the intensity of
the incident light increases.
•When DV is negative, the
current drops.
•When DV is equal to or
more negative than DVs,
the current is zero.

Photoelectric Effect Feature 1
•Dependence of photoelectron kinetic energy on
light intensity
– Classical Prediction
• Electrons should absorb energy continually from the electromagnetic
waves.
• As the light intensity incident on the metal is increased, the electrons
should be ejected with more kinetic energy.
– Experimental Result
• The maximum kinetic energy is independent of light intensity.
• The maximum kinetic energy is proportional to the stopping potential
(DVs).
Section 40.2

•Time interval between incidence of light and
ejection of photoelectrons
• At low light intensities, a measurable time interval should pass between
the instant the light is turned on and the time an electron is ejected from
the metal.
• This time interval is required for the electron to absorb the incident
radiation before it acquires enough energy to escape from the metal.
• Electrons are emitted almost instantaneously, even at very low light
intensities.
Section 40.2

•Dependence of ejection of electrons on light
frequency
• Electrons should be ejected at any frequency as long as the light intensity
is high enough.
• No electrons are emitted if the incident light falls below some cutoff
frequency, ƒc.
• The cutoff frequency is characteristic of the material being illuminated.
• No electrons are ejected below the cutoff frequency regardless of
intensity.
Section 40.2

•Dependence of photoelectron kinetic energy on
light frequency
• There should be no relationship between the frequency
of the light and the electric kinetic energy.
• The kinetic energy should be related to the intensity of
the light.
• The maximum kinetic energy of the photoelectrons
increases with increasing light frequency.
Section 40.2

Photoelectric Effect Features,
Summary
•The experimental results contradict all four
classical predictions.
•Einstein extended Planck’s concept of quantization
to electromagnetic waves.
•All electromagnetic radiation of frequency ƒ from
any source can be considered a stream of quanta,
now called photons.
•Each photon has an energy E and moves at the
speed of light in a vacuum.
– E = hƒ
•A photon of incident light gives all its energy to a
single electron in the metal.
Section 40.2

Photoelectric Effect, Work Function
•Electrons ejected from the surface of the metal
and not making collisions with other metal
atoms before escaping possess the maximum
kinetic energy Kmax.
•Kmax = hƒ – φ
– φ is called the work function of the metal.
– The work function represents the minimum
energy with which an electron is bound in the
metal.
Section 40.2

Does light consist of particles or waves? When one focuses upon the different types of phenomena
observed with light, a strong case can be built for a wave picture:
Phenomenon
Can be explained in terms of
waves.
Can be explained in terms of
particles.
Most commonly observed phenomena with light can be explained by waves. But the photoelectric
effect and the Compton scatering suggested a particle nature for light. Then electrons too were
found to exhibit dual natures.
Wave-Particle Duality: Light
Reflection
Refraction
Interference
Diffraction
Polarization
Photoelectric effect
Compton scattering

While the Bohr model was a major step toward understanding the quantum theory of
the atom, it is not in fact a correct description of the nature of electron orbits. Some
of the shortcomings of the model are:
1. It fails to provide any understanding of why certain spectral lines are brighter than
others. There is no mechanism for the calculation of transition probabilities.
2. The Bohr model treats the electron as if it were a miniature planet, with definite
radius and momentum. This is in direct violation of the uncertainty principle which
dictates that position and momentum cannot be simultaneously determined.
The Bohr model gives us a basic conceptual model of electrons orbits and energies.
The precise details of spectra and charge distribution must be left to quantum
mechanical calculations, as with the Schrödinger equation.
Failures of the Bohr Model (old quantum theory)

Movement of the Electron around the
Nucleus

Schrodinger wave equation is a mathematical
expression describing the energy and position
of the electron in space and time, taking into
account the matter wave nature of the
electron inside an atom.
It is based on three considerations. They are;
• Classical plane wave equation,
• Broglie’s Hypothesis of matter-wave, and
• Conservation of Energy.

What is a wave function?
Wave function is used to describe ‘matter waves’. Matter waves are
very small particles in motion having a wave nature – dual nature of
particle and wave. Any variable property that makes up the matter
waves is a wave function of the matter-wave. Wave function is denoted
by a symbol ‘Ψ’.
Amplitude, a property of a wave, is measured by following the
movement of the particle with its Cartesian coordinates with respect of
time. The amplitude of a wave is a wave function. The wave nature and
the amplitudes are a function of coordinates and time, such that,
Wave function Amplitude = Ψ = Ψ(r,t); where, ‘r’ is the position of the
particle in terms of x, y, z directions.

What is the physical significance of Schrodinger wave
function?
Bohr concept of an atom is simple. But it cannot explain the
presence of multiple orbitals and the fine spectrum arising
out of them. It is applicable only to the one-electron system.
Schrodinger wave function has multiple unique solutions
representing characteristic radius, energy, amplitude.
Probability density of the electron calculated from the wave
function shows multiple orbitals with unique energy and
distribution in space.
Schrodinger equation could explain the presence of multiple
orbitals and the fine spectrum arising out of all atoms, not
necessarily hydrogen-like atoms.

Displacement of a particle from its mean position is given by a simple equation from wave
mechanics, as
y = A⋅sin (ωt − δ)
Quantum mechanically, this particle, describing S.H.M. in accordance with equation has a wave,
or in particular - a matter wave, associated with it, which is represented by a wave function
ψ(x,t). This wave function is not a directly measurable quantity and may be complex in nature.
The wave function associated with a particle moving along +x direction is given by
where, A is the amplitude of oscillations, ω is angular frequency, t is the time, x is position
and v is its velocity.

As ω = 2πν and v = νλ, equation
If ν is the frequency of oscillations, the total energy is given by
E = hν = 2πhν
------2

Therefore the equation 2 becomes
--------4

The electrons are more likely to be found:
(1) in the region a and b
(2) in the region a and c
(3) only in the region c
(4) only in the region a

Basics of Quantum Mechanics
- First Postulate of Quantum
Mechanics -
Quantum physicists are interested in all kinds of physical systems
(photons, conduction electrons in metals and semiconductors, atoms,
etc.). State of these rather diverse systems are represented by the same
type of functions  STATE FUNCTIONS.
First postulate of Quantum mechanics:
Every physically-realizable state of the system is described in quantum
mechanics by a state function  that contains all accessible physical
information about the system in that state.
– Physically realizable states  states that can be studied in laboratory
– Accesible information  the information we can extract from the
wavefunction
– State function  function of position, momentum, energy that is spatially
localized.

- First Postulate of Quantum
Mechanics -
If 1 and 2 represent two physically-realizable states of the system, then
the linear combination
where c1 and c2 are arbitrary complex constants, represents a third
physically realizable state of the system.
Note:
Wavefunction (x,t)  position and time probability amplitude
Quantum mechanics describes the outcome of an ensemble of
measurements, where an ensemble of measurements consists of a very
large number of identical experiments performed on identical non-
interacting systems, all of which have been identically prepared so as to be
in the same state.
2
2
1
1 



 c
c

- Second Postulate of Quantum
Mechanics -
If a system is in a quantum state represented by a wavefunction , then
is the probability that in a position measurement at time t the particle will
be detected in the infinitesimal volume dV.
Note:
 position and time probability density
The importance of normalization follows from the Born interpretation of
the state function as a position probability amplitude. According to the
second postulate of quantum mechanics, the integrated probability
density can be interpreted as a probability that in a position measurement
at time t, we will find the particle anywhere in space.
dV
PdV
2


2
)
,
( t
x


- Second Postulate of Quantum
Mechanics -
Therefore, the normalization condition for the
wavefunction is:
Limitations on the wavefunction:
– Only normalizable functions can represent a quantum
state and these are called physically admissible functions.
– State function must be continuous and single valued
function.
– State function must be a smoothly-varying function
(continuous derivative).
1
)
,
,
(
)
,
,
(
)
,
,
( *
2

 


 

 dV
z
y
x
z
y
x
dV
z
y
x
PdV

- Third Postulate of Quantum
Mechanics -
Every observable in quantum mechanics is represented by an operator which is used to
obtain physical information about the observable from the state function. For an
observable that is represented in classical physics by a function Q(x,p), the corresponding
operator is )
,
( p
x
Q


.
Observable Operator
Position x

Momentum
x
i
p





Energy
)
(
2
)
(
2 2
2
2
2
x
V
x
m
x
V
m
p
E 









Third Postulate:

- More on Operators -
 An operator is an instruction, a symbol which tells us to perform one or more
mathematical acts on a function, say f(x). The essential point is that they act on a
function.
 Operators act on everything to the right, unless the action is constrained by brackets.
 Addition and subtraction rule for operators:
  )
(
)
(
)
( 2
1
2
1 x
f
Q
x
f
Q
x
f
Q
Q







 The product of two operators implies succesive operation:
 
)
(
)
( 2
1
2
1 x
f
Q
Q
x
f
Q
Q





 The product of two operators is a third operator:
2
1
3 Q
Q
Q




 Two operators commute if they obey the simple operator expression:
  1
2
2
1
1
2
2
1
2
1 0
, Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
















- More on Operators -
The requirement for two operators to be commuting
operators is a very important one in quantum mechanics and
it means that we can simultaneously measure the observables
represented with these two operators. The non-commutivity
of the position and the momentum operators (the inability to
simultaneously determine particles position and its
momentum) is represented with the Heisenberg uncertainty
principle, which in mathematical form is expressed as:
and can be generalized for any pair of observables.
 
p
x
p
x



,
2
1
2


D

D

- Fourth Postulate of Quantum Mechanics -
1926 Erwin Schrödinger proposed an equation that describes the evolution of a quantum-
mechanical system  SWE which represents quantum equations of motion, and is of the
form:
t
i
t
x
x
V
x
m
t
x
x
V
x
m 






















 


)
,
(
)
(
2
)
,
(
)
(
2 2
2
2
2
2
2
This work of Schrödinger was stimulated by a 1925 paper by Einstein on the quantum
theory of ideal gas, and the de Broglie theory of matter waves.
Note:
Examining the time-dependent SWE, one can also define the following operator for the
total energy:
t
i
E


 

Fourth (Fundamental) postulate of Quantum mechanics:
The time development of the state functions of an isolated quantum system is governed
by the time-dependent SWE t
i
H 



 /


, where V
T
H




 is the Hamiltonian of the
system.
Note on isolated system:
The TDSWE describes the evolution of a state provided that no observations are made.
An observation alters the state of the observed system, and as it is, the TDSWE can not
describe such changes.

- Fourth Postulate of Quantum
Mechanics -
Examining the time-dependent SWE, one can also define the following operator for the
total energy:
t
i
E


 

Fourth (Fundamental) postulate of Quantum mechanics:
The time development of the state functions of an isolated quantum system is governed
by the time-dependent SWE t
i
H 



 /


, where V
T
H




 is the Hamiltonian of the
system.
Note on isolated system:
The TDSWE describes the evolution of a state provided that no observations are made.
An observation alters the state of the observed system, and as it is, the TDSWE can not
describe such changes.

The position and momentum of a particle cannot be simultaneously measured with
arbitrarily high precision. There is a minimum for the product of the uncertainties of these
two measurements. There is likewise a minimum for the product of the uncertainties of
the energy and time
The Uncertainty Principle

By Planck’s law E = hc/λ, a photon with a short
wavelength has a large energy
Thus, it would impart a large ‘kick’ to the electron
But to determine its momentum accurately,
electron must only be given a small kick
Use light with short wavelength:
accurate measurement of position but not
momentum.
Use light with long wavelength:
accurate measurement of momentum but not
position.

Example
What is a quantum computer?
 A quantum computer is a machine that
performs calculations based on the laws
of quantum mechanics, which is the
behavior of particles at the sub-atomic
level.

Representation of Data - Qubits
A bit of data is represented by a single atom that is in one of
two states denoted by |0> and |1>. A single bit of this form is
known as a qubit
A physical implementation of a qubit could use the two
energy levels of an atom. An excited state representing |1>
and a ground state representing |0>.
Excited
State
Ground
State
Nucleus
Light pulse of
frequency l for
time interval t
Electron
State |0> State |1>

V(x)=0 for L>x>0
V(x)=∞ for x≥L, x≤0
Particle in a 1-Dimensional Box






E
x
V
dx
d
m
)
(
2 2
2
2

Classical Physics: The particle can
exist anywhere in the box and follow
a path in accordance to Newton’s
Laws.Quantum Physics: The particle
is expressed by a wave function and
there are certain areas more likely
to contain the particle within the
box.



E
x
V
dx
x
d
m



)
(
)
(
2 2
2
2

KE PE TE
Time independent Schrödinger Equation
)
(
)
(
)
,
( x
t
f
t
x 


Wave function is dependent on time and position
function:
Time Independent Schrödinger Equation
Applying boundary conditions:



E
dx
x
d
m




*
)
(
2 2
2
2

Region I and III:


E
dx
x
d
m


2
2
2
)
(
2

Region II:
0
2


V(x)=0 V(x)=∞
V(x)=∞
0 L x
Region I Region II Region III

Finding the Wave Function


E
dx
x
d
m


2
2
2
)
(
2


 2
2
2
)
(
k
dx
x
d




E
m
dx
x
d
2
2
2
2
)
(



This is similar to the general differential equation:
2
2 2

mE
k 
m
k
E
2
2
2


kx
B
kx
A cos
sin 


So we can start applying boundary conditions:
x=0 ψ=0
k
B
k
A 0
cos
0
sin
0 
 0
1
*
0
0 


 B
B
x=L ψ=0
0

A
kL
Asin
0  
n
kL 
L
x
n
A
II

 sin

2
2
2
4
2 
m
h
k
E 

2
h


2
2
2
2
2
4
2 

m
h
L
n
E  2
2
2
8mL
h
n
E 
Our new wave function:
But what is ‘A’?
Calculating Energy Levels:
Normalizing wave function:
1
)
sin
(
0
2


L
dx
kx
A
1
4
2
sin
2 0
2








L
k
kx
x
A
1
4
2
sin
2
2












L
n
L
L
n
L
A


Since n= *
1
2
2






 L
A
L
A
2

Our normalized wave function is:
L
x
n
L
II

 sin
2


E
x/L x/L
E
Particle in a 1-Dimensional Box
n=1
n=2
n=3
n=4
n=1
n=2
n=3
n=4
L
x
n
L
II

 sin
2

2
2
sin
2







L
x
n
L
II


Applying the
Born Interpretation

• Now we consider the situation where classically the particle does not have enough
energy to surmount the potential barrier, E < V0.
• The quantum mechanical result, however, is one of the most remarkable features of
modern physics, and there is ample experimental proof of its existence. There is a small,
but finite, probability that the particle can penetrate the barrier and even emerge on the
other side.
• The wave function in region II becomes
76
Tunneling

Physics Module 2 .pdf on quantum physics

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