# Wave-Particle Duality

A fundamental distinction was drawn in classical physics between waves and particles. This was refuted by experimental evidence that showed waves demonstrating particle-like character and vice versa.

Evidence for the particle-like character of electromagnetic radiation was given by the photoelectric effect. This is the name given to the phenomenon whereby electrons are ejected from a sample of metal that is exposed to ultraviolet radiation. There are three important experimental observation about this effect:

1. No electrons are ejected, regardless of the intensity of the radiation, unless its frequency is greater than some critical value, ν0, which is characteristic of the metal.
2. The maximum kinetic energy of the emitted electrons is proportional to the frequency of the incident radiation, but independent of the intensity of this radiation.
3. If the frequency is greater than ν0, then emission of electrons occurs immediately after absorption of radiation, even at low light intensities. There is no delay between absorption and emission.

These effects are difficult to rationalise if the incident radiation is treated as a wave. Particularly puzzling is the apparent lack of effect of the intensity of the radiation.

The observations are most easily explained by considering the radiation to be composed of a stream of particles, each of energy hν. These particles are known as photons.

From this we can state that ejection of an electron would occur when a photon collides with one on the surface of the metal, giving up all its energy to the electron and allowing it to escape from its bound state. By the conservation of energy, we may write the kinetic energy of the emitted electrons as: where Φ is the work function of the metal, a characteristic value of a given metal which is the binding energy of the outermost electrons in that metal.

We can see that it is necessary for the energy of the incoming photons to exceed the work function for photoejection to occur. (If the incoming photons have less energy than this, then the electrons will receive insufficient energy from them to allow them to overcome their binding energy and be emitted.) The work function may be expressed in terms of a frequency as 0, and since we have stated that the incoming electrons must have an energy greater than this, we have explained the first experimental observation.

From the above equation, we can clearly see that this theory agrees with the second experimental observation, that the kinetic energy of the emitted electrons is directly proportional to the frequency ν of the incident radiation.

The intensity of the radiation would measure the number of photons hitting a given surface area per unit time. Though this might affect the rate of electron emission (the number of electrons emitted per unit time), it does not affect the kinetic energy with which they are emitted.

Ejection of electrons is immediate because the photons collide with the electrons, instantaneously giving up all their energy and thus immediately increasing the energy of the electron to the point where it will be emitted.

The wave-like properties of particles are most readily demonstrated by the fact that electrons (and other particles such as α particles, neutrons and even hydrogen molecules) can be diffracted, which is a characteristic property of waves, arising as it does from interference between peaks and troughs in the waves.

The fundamental link between particles and waves is embodied in the de Broglie relation:

This suggests that all particles with a linear momentum, p, have an associated wavelength, λ.

The reason this relationship was not a part of classical physics is that macroscopic objects, those which are observed on a daily basis, have such large linear momenta (due to their large masses) that their wavelengths are too short to be observed. Only with very light particles such as the electron can the relationship be demonstrated experimentally.