Review of Classical Wave

Week 2 Principles of Quantum Mechanics
Energy Quanta
Wave-Particle Duality
Review of Classical Waves
The double-slit experiment
The Heisenberg Uncertainty Principle
Schrodinger’s Wave Equation

Planck’s Energy Quanta: Photoelectric Effects
• The photoelectric effect shows the discrete nature of the photon and
demonstrates the particle-like behavior of the photon .
• Einstein explained the photoelectric effect by postulating that, in a
photon, the energy was proportional to the frequency (1905).

Compton Effect
• In this experiment, an x-ray beam is incident on a solid. A portion of the
beam was deflected and the frequency of the deflected wave was shifted
compared to the incident wave. The Compton effect, was explained by
proposing that photons carry momentum.
• If a photon collides with an
electron, the wavelength and
trajectory of the photon is
observed to change. The
wavelength shift of light after
collision with an electron is
consistent with a transfer of
momentum from a photon to
the electron. The loss of
photon energy is reflected in a
red shift of its frequency.

De Broglie’s Wave-Particle Duality Principles
• In 1924. de Broglie postulated the existence of matter waves. He suggested
that since waves exhibit particle-like behavior, then particles should also be
expected to show wave-like properties.
• The hypothesis of de Broglie was the existence of a wave-particle duality
principle. The momentum of a photon is given by
• Then, de Broglie hypothesized that the wavelength of a particle can be
expressed as (de Broglie wavelength of the matter wave).

Davisson-Germer Experiment

Electron Diffraction through Double Slits
• Classically, we would predict that electrons passing through slits in a screen
should continue in straight lines, forming an exact image of the slits on the rear
screen. In practice, however, a series of lines is formed on the rear screen,
suggesting that the electrons have been somehow deflected by the slits.

Review of Classical Wave
• A wave is a periodic oscillation. It is convenient to describe waves using
complex numbers. For example consider the function
where x is position. This function can be plotted on the complex plane as
a function of position, x. The phase of the function is the angle on the
complex plane.

Review of Classical Wave
• The wavelength is defined as the distance between spatial repetitions of
the oscillation. This corresponds to a phase change of
• This wave is independent of time, and is known as a standing wave.
• We could define a function whose phase varies with time:
• We define the period, T, as the time between repetitions of the oscillation

Review of Classical Wave
• We can combine time and spatial phase oscillations to make a traveling
wave.
• The intensity of this wave is uniform everywhere it is known as a
plane wave.
• A plane wave has at least four dimensions (real amplitude, imaginary
amplitude, x, and t), so we plot planes of a given phase. These plane waves
move through space at the phase velocity of the wave

The Double Slit Experiment
• Far from the double slit, the electrons from each slit can be described by
plane waves, where s is the separation between the slits and L is the
distance to the viewing screen. When the planes of constant phase collide,
a bright line corresponding to a high intensity of electrons is observed.

• At the viewing screen we have

The Double Slit Experiment
• At the screen, constructive interference between the plane waves from
each slit yields a regular array of bright lines, corresponding to a high
intensity of electrons. In between each pair of bright lines, is a dark band
where the plane waves interfere destructively, i.e. the waves are out of
phase with one another.
• The spacing between the bright lines at the viewing screen is
• It is notable that the fringe pattern is independent of intensity. Thus, the
interference effect should be observed even if just a single electron is
fired at the slits at a time.
• The only conclusion is that the electron – which we are used to thinking
of as a particle - also has wave properties.

• The cumulative electron distribution after passage through a double slit.
Just a single electron is present in the apparatus at any time.
A. Tanamura et al., Am. J. Phys.
57, 117 (1989).

The Electromagnetic Spectrum

Heisenberg’s Uncertainty Principle
• Heisenberg’s uncertainty principle, given in 1927, applies primarily to very
small particles, and states that we cannot describe with absolute accuracy
the behavior of these subatomic particles.
• The uncertainty principle describe a fundamental relationship between
conjugate variables, including position and momentum and also
energy and time.
• The first statement of the uncertainty principle is that it is impossible to
simultaneously describe with absolute accuracy the position and
momentum of a particle.
• The second statement of the uncertainty principle is that it is impossible to
simultaneously describe with absolute accuracy the energy of a particle
and the instant of time the particle has this energy.

Heisenberg’s Uncertainty Principle
• The uncertainty principle is not very relevant to everyday objects

Schrodinger’s Wave Equation
• The various experimental results involving electromagnetic waves and
particles, which could not be explained by classical laws of physics, showed
that a revised formulation of mechanics was required.
• Erwin Schrodinger (1926), provided a formulation, called wave mechanics,
which incorporated the principles of quanta introduced by Planck, and the
wave-particle duality principle introduced by de Broglie.
• Based on the wave-particle duality principle. we will describe the motion of
electrons in a solid or nanostructure by wave theory. This wave theory is
described by Schrodinger's wave equation.