For centuries, scientists debated whether light is a wave or a particle. By 1900, the wave theory seemed complete, explaining interference, diffraction, and polarization. But experiments on blackbody radiation and the photoelectric effect forced physics to accept that light also behaves as a stream of particles, or photons, each carrying energy E = hf. This chapter introduces this dual nature of light.
In 1905, Einstein explained the photoelectric effect using the photon picture, for which he received the Nobel Prize. In 1924, Louis de Broglie proposed that matter also has a wave nature, with every particle associated with a wavelength lambda = h/p. This profound symmetry - waves behave like particles and particles like waves - is the wave-particle duality.
The chapter studies the photoelectric effect in detail, photon properties, and the de Broglie hypothesis confirmed by the Davisson-Germer experiment. These ideas are the foundation of quantum mechanics and explain the behaviour of atoms, electrons, and light.
The photoelectric effect is the emission of electrons from a metal surface when light of suitable frequency falls on it. The emitted electrons are called photoelectrons. The effect was discovered by Heinrich Hertz in 1887, who noticed that ultraviolet light falling on the cathode of a discharge tube helped the discharge.
The classical wave theory could not explain three key observations: 1. The maximum kinetic energy of the emitted electrons is independent of the intensity of light, but depends on its frequency. 2. No electrons are emitted if the frequency of light is below a certain threshold frequency, no matter how intense the light. 3. Emission is instantaneous, with no measurable time lag, even for very weak light.
These observations contradicted the wave picture, where energy is spread smoothly and a time lag should exist, and higher intensity should give more energy per electron.
Einstein explained the photoelectric effect by postulating that light consists of photons, each carrying energy:
E = h f
where h = 6.63 x 10^-34 J s is Planck's constant and f the frequency. A photon can transfer all its energy to a single electron in the metal. Part of this energy, the work function phi_0, is used to free the electron from the metal surface, and the rest appears as the kinetic energy of the emitted electron. Einstein's photoelectric equation is:
K_max = h f - phi_0
The work function is phi_0 = h f0, where f0 is the threshold frequency. The maximum kinetic energy is:
K_max = h (f - f0) = e V0
where V0 is the stopping potential. The stopping potential is the minimum retarding potential that just stops the most energetic photoelectrons. A graph of stopping potential versus frequency is a straight line with slope h/e, from which Planck's constant can be measured.
From the photoelectric equation and experiment, the following laws hold: 1. For a given metal and frequency, the number of photoelectrons emitted per second is proportional to the intensity of the incident light. 2. For a given metal, there exists a threshold frequency below which no emission occurs, whatever the intensity. 3. The maximum kinetic energy of the photoelectrons increases linearly with the frequency of the incident light and is independent of intensity. 4. The photoelectric emission is instantaneous; the time lag is less than 10^-9 seconds.
The intensity of light determines the number of photons per second, and hence the photocurrent, while the frequency determines the energy of each photon and the kinetic energy of the electrons.
A photon is a quantum of electromagnetic energy with the following properties: 1. The energy of a photon is E = h f = h c / lambda. 2. The momentum of a photon is p = E/c = h/lambda. 3. A photon has zero rest mass and always travels at the speed of light. 4. The total energy of a beam is the sum of the energies of its photons; photon energy is not divisible.
The photon picture explains why the energy of photoelectrons is independent of intensity: each electron absorbs one photon, and the photon energy is set by the frequency. The momentum of photons leads to radiation pressure, and the idea that energy is exchanged in discrete quanta is fundamental to quantum physics.
Louis de Broglie proposed in 1924 that if light, a wave, can behave like a particle, then particles such as electrons should also behave like waves. He associated a wavelength with every moving particle:
lambda = h / p = h / (m v)
This is the de Broglie wavelength. For an electron accelerated through a potential difference V, the de Broglie wavelength is:
lambda = h / sqrt(2 m e V) = 12.27 / sqrt(V) angstrom
The de Broglie wavelength of macroscopic objects is extremely small (for a cricket ball it is about 10^-34 m), which is why we never observe the wave nature of everyday objects. For electrons, however, the wavelength is comparable to atomic spacings and X-ray wavelengths, so their wave nature is observable.
The wave nature of electrons was confirmed experimentally in 1927 by Clinton Davisson and Lester Germer. They directed a beam of electrons at a nickel crystal and observed that the electrons were diffracted, producing intensity maxima at specific angles, just as X-rays are diffracted by a crystal.
The diffraction pattern could be explained only if the electrons had a wavelength given by the de Broglie formula lambda = h/sqrt(2meV). The accelerating potential at which maximum scattering occurred matched the predicted wavelength. This experiment conclusively demonstrated the wave nature of matter.
The dual nature is now established: electromagnetic radiation shows wave-like behaviour in interference and diffraction and particle-like behaviour in the photoelectric and Compton effects, while electrons show particle-like behaviour in the laboratory and wave-like behaviour in diffraction experiments.
| Quantity | Formula | Value/Remark |
|---|---|---|
| Photon energy | E = h f = h c/lambda | h = 6.63 x 10^-34 J s |
| Photon momentum | p = h/lambda | Zero rest mass |
| Work function | phi_0 = h f0 | Metal property |
| Photoelectric equation | K_max = h f - phi_0 | Einstein 1905 |
| Stopping potential | e V0 = K_max | Slope h/e vs f |
| de Broglie wavelength | lambda = h/m v | For all particles |
| Electron wavelength | lambda = 12.27/sqrt(V) angstrom | Accelerated electron |
| Threshold frequency | f0 = phi_0/h | No emission below |
| Observation | Wave prediction | Actual result |
|---|---|---|
| K_max vs intensity | Should increase | Independent |
| K_max vs frequency | No dependence | Linear increase |
| Threshold frequency | None | Exists |
| Time lag | Should exist | Negligible |
This chapter established the dual nature of radiation and matter. The photoelectric effect, unexplained by wave theory, was explained by Einstein with photons of energy E = hf, giving K_max = hf - phi_0. The laws of photoelectric emission linked intensity to photocurrent and frequency to electron energy. Photons carry energy and momentum p = h/lambda but have zero rest mass. De Broglie extended the idea to matter with lambda = h/mv, and the Davisson-Germer experiment confirmed electron diffraction. Together these results reveal that light and matter each show both wave and particle behaviour - the wave-particle duality that lies at the heart of quantum mechanics.