Photoelectric Effect

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

Another contributing factor to the emergence of the theory of quantum mechanics came with the realization of the particle nature of light through explanation of the photoelectric effect.

Consider a system composed of light hitting a metal plate. Experiments, such as that by Hertz in 1887, and one performed by Hallwachs, Stoletov, and Lenard in 1900, showed that a current can be measured when light is incident on the metal plate. At that time, the (classical) point of view was that an electron was bound inside of an atom, and thus energy was needed in order to release it from the atom. This energy could be supplied in the form of light. The classical point of view also included the idea that the energy of the light was proportional to its intensity. Therefore, if enough energy is absorbed by the electron, then it will eventually be released. However, this was not the case. Several odd results came from these studies. First of all, it was noted that, while the current did appear to be proportional to the intensity of the incident light, there was a certain minimum frequency of light below which no current would be produced, regardless of the intensity of the incident beam. Also, the stopping potential of the emitted electrons appeared to depend upon the frequency of the radiation, and not on the intensity at all. Finally, the emission appeared to take place instantaneously for any intensity so long as the minimum frequency condition was satisfied.

In 1905, Einstein offered a possible explanation for these odd observations. Einstein realized that the classical view of light as a wave was not entirely true, that light must also behave like a particle. This allowed him to postulate that the energy of the incident radiation was not continuous, but was rather composed of quantized packets, proportional to the frequency of the wavelength of incident light, as Planck did in his explanation of the spectral distribution of black-body radiation. These corpuscles could then be completely absorbed by an atom, rather then spreading out over the structure as a wave would, so that the absorption and subsequent emission would happen instantly. He commented that since electrons were inherently bound to the atom, a certain minimum energy would be required to remove them, and thus, if a corpuscle did not have enough energy (i.e. its frequency was too low), then the atom would merely absorb and release it, rather then emitting an electron as well. From this result, Millikan was able to confirm Einstein's theory a few years later by showing that the stopping potential did indeed depend linearly on the frequency, with an additive term corresponding to the minimum energy required to remove the electron, or the work function of the metal.

The equation describing the kinetic energy of the emitted electron is:

\frac{1}{2}mv^2 = h\nu - W

Where W is the work function and \nu\! is the frequency of the incident photon.

From these results, it was clear that light can behave in a particle-like manner. However, the existence of various interference and diffraction experiments still gave evidence for a wave-like nature as well, and thus the dual nature of light was exposed, in stark contrast to classical physics.

Problem

(Ronald Gautreau, Theory and Problems of Modern Physics, Problem 9.13)

The emitter in a photoelectric tube has a threshold wavelength of 6000 Å. Determine the wavelength of the light incident on the tube if the stopping potential for this light is 2.5 V.

Solution

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