Introduction To Quantum Mechanics - Photons: The Quantisation of Light - The Photoelectric Effect

The Photoelectric Effect

In 1887 Heinrich Hertz observed that light can eject electrons from metal. In 1902 Philipp Lenard discovered that the maximum possible energy of an ejected electron is related to the frequency of the light, not to its intensity; if the frequency is too low, no electrons are ejected regardless of the intensity. The lowest frequency of light that causes electrons to be emitted, called the threshold frequency, is different for every metal. This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the intensity of the radiation.

Einstein explained the effect by postulating that a beam of light is a stream of particles (photons), and that if the beam is of frequency f then each photon has an energy equal to hf. An electron is likely to be struck only by a single photon, which imparts at most an energy hf to the electron. Therefore, the intensity of the beam has no effect; only its frequency determines the maximum energy that can be imparted to the electron.

To explain the threshold effect, Einstein argued that it takes a certain amount of energy, called the work function, denoted by φ, to remove an electron from the metal. This amount of energy is different for each metal. If the energy of the photon is less than the work function then it does not carry sufficient energy to remove the electron from the metal. The threshold frequency, f₀, is the frequency of a photon whose energy is equal to the work function:

If f is greater than f₀, the energy hf is enough to remove an electron. The ejected electron has a kinetic energy E_K which is, at most, equal to the photon's energy minus the energy needed to dislodge the electron from the metal:

Einstein's description of light as being composed of particles extended Planck's notion of quantised energy: a single photon of a given frequency f delivers an invariant amount of energy hf. In other words, individual photons can deliver more or less energy, but only depending on their frequencies. However, although the photon is a particle it was still being described as having the wave-like property of frequency. Once again, the particle account of light was being "compromised".

The relationship between the frequency of electromagnetic radiation and the energy of each individual photon is why ultraviolet light can cause sunburn, but visible or infrared light cannot. A photon of ultraviolet light will deliver a high amount of energy—enough to contribute to cellular damage such as occurs in a sunburn. A photon of infrared light will deliver a lower amount of energy—only enough to warm one's skin. So an infrared lamp can warm a large surface, perhaps large enough to keep people comfortable in a cold room, but it cannot give anyone a sunburn.

If each individual photon had identical energy, it would not be correct to talk of a "high energy" photon. Light of high frequency could carry more energy only because of flooding a surface with more photons arriving per second. Light of low frequency could carry more energy only for the same reason. If it were true that all photons carry the same energy, then if you doubled the rate of photon delivery, you would double the number of energy units arriving each second. Einstein rejected that wave-dependent classical approach in favour of a particle-based analysis where the energy of the particle must be absolute and varies with frequency in discrete steps (i.e. is quantised). All photons of the same frequency have identical energy, and all photons of different frequencies have proportionally different energies.

In nature, single photons are rarely encountered. The sun emits photons continuously at all electromagnetic frequencies, so they appear to propagate as a continuous wave, not as discrete units. The emission sources available to Hertz and Lennard in the 19th century shared that characteristic. A sun that radiates red light, or a piece of iron in a forge that glows red, may both be said to contain a great deal of energy. It might be surmised that adding continuously to the total energy of some radiating body would make it radiate red light, orange light, yellow light, green light, blue light, violet light, and so on in that order. But that is not so for otherwise larger suns and larger pieces of iron in a forge would glow with colours more toward the violet end of the spectrum. To change the color of such a radiating body it is necessary to change its temperature, and increasing its temperature changes the quanta of energy that are available to excite individual atoms to higher levels and permit them to emit photons of higher frequencies. The total energy emitted per unit of time by a sun or by a piece of iron in a forge depends on both the number of photons emitted per unit of time and also on the amount of energy carried by each of the photons involved. In other words, the characteristic frequency of a radiating body is dependent on its temperature. When physicists were looking only at beams of light containing huge numbers of individual and virtually indistinguishable photons it was difficult to understand the importance of the energy levels of individual photons. So when physicists first discovered devices exhibiting the photoelectric effect, the effect that makes the light meters of modern cameras work, they initially expected that a higher intensity of light would produce a higher voltage from the photoelectric device. They discovered that strong beams of light toward the red end of the spectrum might produce no electrical potential at all, and that weak beams of light toward the violet end of the spectrum would produce higher and higher voltages. Einstein's idea that individual units of light may contain different amounts of energy depending on their frequency made it possible to explain the experimental results that hitherto had seemed quite counter-intuitive.

Although the energy imparted by photons is invariant at any given frequency, the initial energy-state of the electrons in a photoelectric device prior to absorption of light is not necessarily uniform. Therefore anomalous results may occur in the case of individual electrons. An electron that was already excited above the equilibrium level of the photoelectric device might be ejected when it absorbed uncharacteristically low frequency illumination. Statistically, however, the characteristic behavior of a photoelectric device will reflect the behavior of the vast majority of its electrons, which will be at their equilibrium level. This point is helpful in comprehending the distinction between the study of individual particles in quantum dynamics and the study of massed particles in classical physics.