Quantised Light and the Photoelectric Effect

When a wave description is not enough

Wave behaviour explains phenomena such as interference and diffraction. Interference is the superposition of waves, producing reinforcement in some places and cancellation in others. Diffraction is the spreading or bending associated with waves passing an opening or obstacle. These observations firmly establish a wave aspect of light.

However, classical nineteenth-century physics could not account satisfactorily for several measured results: the distribution of radiation from hot bodies, photoelectric emission, the temperature dependence of the heat capacity of solids, and atomic line spectra. Rather than discard observations, scientists needed a new rule for how matter exchanges energy with radiation. We develop that rule here and use it for spectra in the next lesson.

Hot objects and the black-body pattern

A heated object emits radiation over a range of wavelengths. At modest temperatures much of it is infrared. As the temperature rises, visible emission becomes noticeable, first red and then with increasing shorter-wavelength contributions. An ordinary surface can reflect, absorb, and transmit different fractions of incident radiation, making its behaviour depend on its material.

A black body is an ideal perfect absorber of incident radiation at every frequency. At a given temperature its emitted spectrum is determined by temperature, rather than the identity of a surface material. A small hole leading into an otherwise closed cavity is a useful approximation: radiation entering the hole undergoes repeated internal reflections and is very likely absorbed.

Perfect absorption at all frequencies does not mean equal emission intensity at all wavelengths. At each temperature the emitted intensity rises to a maximum and then falls. Raising temperature increases emission and shifts the peak towards shorter wavelength. Any successful theory has to explain the whole curve and its temperature dependence.

Planck’s discrete energy exchange

Planck proposed that the oscillators associated with matter emit and absorb energy in definite packets. For radiation of frequency $\nu$, the smallest packet is a quantum of energy $h\nu$. The allowed exchanges are whole-number multiples of this amount. A staircase is a useful analogy: a person can stand on a step, not on a height halfway between adjacent steps.

The step size is not universal for every frequency. Higher-frequency radiation has a larger energy quantum. This is why the quantum idea concerns both discreteness and the relationship between energy and frequency. In describing light as particles, we call its energy packets photons.

For a collection of photons at one frequency, multiply the single-photon energy by their number. One mole contains $N_A$ photons, so its energy is $N_{A}h\nu$. Power is energy per second. If all the power $P$ leaves as monochromatic radiation, dividing by the energy per photon gives the number emitted per second.

What the photoelectric experiment reveals

When sufficiently high-frequency light strikes a suitable clean metal surface, electrons are emitted. Three observations require explanation. Emission begins without a measurable classical waiting time; below a characteristic threshold frequency no electrons emerge even under very intense illumination; above threshold, increasing frequency increases the maximum kinetic energy of the emitted electrons.

Above threshold, increasing intensity at a fixed frequency increases the number of electrons emitted, provided comparable experimental conditions are maintained. It does not increase the maximum kinetic energy of each emitted electron. For potassium, for example, very bright red light may fail while weak yellow light succeeds because the relevant difference is photon energy, not simply brightness.

Einstein explained these results by treating the light–electron interaction as the transfer of one photon’s energy to one electron. A minimum energy is needed to free the electron from the metal. Any remaining energy appears as the electron’s kinetic energy. Many electrons can be emitted because a beam contains many photons, not because one photon repeatedly supplies its full energy to many electrons.

The energy balance can now be written precisely. The largest emitted kinetic energy equals the incident photon energy minus the work function; the equivalent threshold-frequency form makes the surplus frequency explicit.

An electronvolt, $\mathrm{eV}$, is an energy unit: $1\mathrm{eV}=1.602176\times 10^{-19}\mathrm{J}$. Do not confuse it with a volt, which measures electric potential difference. If work function and photon energy are initially in electronvolts, they can be subtracted in that unit; convert the resulting kinetic energy to joules before using SI mass to find speed.

Keeping both aspects of radiation

Interference and diffraction demand a wave account; the photon account explains discrete exchanges such as the photoelectric effect. We therefore describe radiation as having wave–particle duality. The useful description depends on the experiment, but light is one physical phenomenon, not two unrelated substances.

A photon is not a tiny classical metal ball. In particular, it has zero rest mass while carrying energy and momentum. These facts prepare us for the surprising result that particles of matter, including electrons, also show wave behaviour.