2. Goals
• To study particles as waves, de Broglie waves
• To consider particles as waves with electron
diffraction and electron microscopy
• To state and explain the difficult conceptual
concept of the Heisenberg uncertainty principle
• To study the general tool for description of a
system—wave functions
• To consider the wide-ranging coverage and
implications of the wave description for an
electron in chemical systems, the Schrödinger
equation
3. Introduction
• At the turn of the 20th century,
Albert Einstein helped lead
science to light as a particle and
wave–particle duality. The next
logical step was not far behind.
De Broglie, Heisenberg, and
eventually Schrödinger
developed a formalism to treat
the particle (an electron) as a
wave spawning the new
adventure, quantum theory.
• The electron micrograph of a
fly’s foot depends on wave-
interference properties of a
fundamental atomic particle,
the electron.
4. De Broglie waves
• If you asked the baseball
pitcher about the wavelength
of his fastball, he’d likely
send you off to deep in the
outfield. But, despite the
seeming conundrum,
macroscopic objects do have
wave properties, and there is
particle–wave duality. The
part that’s so hard to see is
the wavelengths of de
Broglie waves. For fastballs
and rifle bullets, they have
wavelengths smaller than
atomic nuclei by orders of
magnitude. The genesis of
the idea is shown in Figure
39.2 at right.
5. Electron diffraction sets a foundation for microscopy
• Heated filaments and electrostatic lenses can
create and manipulate beams of electrons for
experimentation. Refer to Figure at the bottom of
the slide.
• Graphs of the results for electron scattering are
shown in Figure at right.
6. Probabilities and electron interference
• Electrons flowing through a slit will form interference patterns like those shown in
the Figure.
7. Heisenberg’s Uncertainty Principle (HUP)
Heisenberg stated that you can’t know the position and momentum of a particle
simultaneously. The principle applies in other applications such as spectral-line
width and laser-pulse duration, preventing high-resolution femtosecond
spectroscopy.
•There is HUP between momentum and position along the same space dimension.
•There is also HUP between energy and time.
10. Schrödinger’s wave equation
• The process presumes that a proper operator working on an
equation which rightly describes the potential and kinetic
components of a system will return eigenvalues and
eigenvectors of the system.
• For example, a vibrational Schrödinger equation can be built
upon Hooke’s Law. When the Hamiltonian operator operates on
the wave equation, eigenvalues will come back as the ascending
energies of sequentially higher vibrational levels.
11. Wave packets
• A wave packet can be built by superposition of a large number of similar oscillating
waves with different wave numbers but similar amplitudes. This package of waves
can be set into motion to … circumvent the uncertainty principle. The packet
samples a problem from many different starting points so the outcome will contain
elements of the answer, each from a different wave.
• Fourier analysis is necessary to “unpack the components of a wave packet” to
discern the pieces of the puzzle … one by one. Figure below illustrates a wave
packet.
