0.3 Quantum energy levels in atoms  (Page 4/9)

We can account for these results in a straightforward but perhaps non-obvious manner. (Einstein providedthe explanation in 1905.) Since the kinetic energy of the emitted photoelectrons increases proportionally with increases in thefrequency of the light above the threshold frequency, we can conclude from conservation of total energy that the energy supplied by the lightto the ejected electron must be proportional to its frequency: $(E, \nu )$ . This does not immediately account for the existence of the threshold frequency, though, since it would stillseem to be the case that even low frequency light would possess high energy if the intensity were sufficient. By this reasoning,high intensity, low frequency light should therefore produce as many photoelectrons as are produced by low intensity, highfrequency light. But this is not observed.

This is a very challenging puzzle, and an analogy helps to reveal the subtle answer. Imagine trying to knockpieces out of a wall by throwing objects at it. We discover that, no matter how many ping pong balls we throw, we cannot knock out apiece of the wall. On the other hand, only a single bowling ball is required to accomplish the task. The results of this"experiment" are similar to the observations of the photoelectric effect: very little high frequency light canaccomplish what an enormous amount of low frequency light cannot. The key to understanding our imaginary experiment is knowing that,although there are many more ping-pong balls than bowling balls, it is only the impact of each individual particle with the wall whichdetermines what happens.

Reasoning from this analogy, we must conclude that the energy of the light is supplied in "bundles"or "packets" of constant energy, which we will call photons . We have already concluded that the light supplies energy to the electron which is proportional to the lightfrequency. Now we can say that the energy of each photon is proportional to the frequency of the light. The intensity of thelight is proportional to the number of these packets. This now accounts for the threshold frequency in a straightforward way. Fora photon to dislodge a photoelectron, it must have sufficient energy, by itself, to supply to the electron to overcome itsattraction to the metal. Although increasing the intensity of the light does increase the total energy of the light, it does notincrease the energy of an individual photon. Therefore, if the frequency of the light is too low, the photon energy is too low toeject an electron. Referring back to the analogy, we can say that a single bowling bowl can accomplish what many ping-pong ballscannot, and a single high frequency photon can accomplish what many low frequency photons cannot.

The important conclusion for our purposes is that light energy is quantized into packets of energy . The amount of energy in each photon is given by Einstein’s equation,

where $h$ is a constant called Planck’s constant.

Quantized energy levels in hydrogen atoms

We can combine the observation of the hydrogen atom spectrum with our deduction that light energy is quantizedinto packets to reach an important conclusion. Each frequency of light in the spectrum corresponds to a particular energy of lightand, therefore, to a particular energy loss by a hydrogen atom, since this light energy is quantized into packets. Furthermore, since only certainfrequencies are observed, then only certain energy losses are possible. This is only reasonable if the energy of each hydrogenatom is restricted to certain specific values. If the hydrogen atom could possess any energy, then it could lose any amount of energyand emit a photon of any energy and frequency. But this is not observed. Therefore, the energy of the electron in a hydrogen atommust be restricted to certain energy levels .