0.4 Quantum electron energy levels in an atom  (Page 6/6)

How much energy does a photon contain? This is revealed by looking at [link] (d), which shows that the kinetic energy of the ejected electrons increases in direct proportion to the frequency provided that the frequency is above the threshold. We can conclude that the light supplies energy to the electron, which is proportional to the light frequency, so the energy of each photon is proportional to the frequency of the light. This now accounts for the observation that the frequency of the light source must be above the threshold frequency. For a photon to dislodge a photoelectron, it must have sufficient energy, by itself, to supply to the electron to overcome its attraction to the metal. It does not get any help from other photons, just like a single ping-pong ball acts alone against the wall. Since each photon must have sufficient energy and since the energy is proportional to the frequency, then each photon must be of a sufficient minimum frequency.

Increasing the intensity of the light certainly must increase the total energy of the light, since we observe this in everyday life. This means that the intensity of the light is proportional to the number of photons in the light but not the energy of each individual photon. Therefore, if the frequency of the light is too low, the photon energy is too low to eject an electron. Think again of the analogy: we can say that a single bowling bowl can accomplish what many ping-pong balls cannot, 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 proportional to the frequency of the light. Einstein first provided these conclusions, along with the equation which gives the energy of a photon of frequency ν

E = h ν

where h is a constant called Planck’s constant.

Observation 3: quantum energy levels in hydrogen atoms

Observation 1 showed us that only certain frequencies of light are emitted by hydrogen atoms. Observation 2 showed us that the energy of light is quantized into photons, or packets of energy, whose energy is proportional to the frequency of the light ν . We can now think about combining these two observations into a single observation about the hydrogen atom. When a hydrogen atom emits light, it must be emitting a photon of energy and is therefore losing energy. A hydrogen atom consists only of a nucleus and a single electron moving about that nucleus. The simplest (and perhaps only) way for the hydrogen atom to lose energy is for the electron to lose some of its energy. Therefore, when a hydrogen atom emits radiation of a certain frequency, it is emitting a photon of a specific energy, and therefore, the electron loses that same very specific energy.

In the spectrum of hydrogen, only certain frequencies are emitted. That means that only certain amounts of energy loss are possible for the electron in a hydrogen atom. How can this be? Why can’t an electron in a hydrogen atom lose any amount of energy? The answer becomes clearer by thinking of an analogy, in this case of walking down a staircase or walking down a ramp. When you walk down a ramp, you can change your elevation by any amount you choose. When you walk down a staircase, you can only change your elevation by fixed amounts determined by the fixed heights of the steps and the difference in heights of those steps. The energy of an electron is like the height of each step on a staircase, not like the height on a ramp, since the energy can only be changed by certain specific amounts. This means that the energy of an electron in a hydrogen atom can only be certain specific values, called “energy levels.” In other words, the energy of a hydrogen atom is “quantized.”

The Rydberg equation tells us what these energy levels are. Recall that every frequency emitted by a hydrogen atom is predicted by the simple equation:

$\nu \mathrm{=}R\mathrm{\times }\left(\frac{1}{n^2}\mathrm{-}\frac{1}{m^2}\right)$

Each emitted frequency must correspond to a certain energy h ν , and this energy must be the energy lost by the electron. This energy must therefore be the difference between two electron energy levels in the hydrogen atom. Let’s label the energy the electron starts with as E m , where m is just an index that tells us where the electron starts. Similarly, let’s label the energy the electron finishes with as E _n , where n is just a different index. The electron loses energy equal to E _m – E _n , and this must equal the photon energy emitted:

h ν = E _m - E _n

We should be able to compare these two equations, since both contain a difference between two quantities that depend on two indices, m and n . Each energy of the electron might be given by an index n as

$E_n\mathrm{=}\mathrm{-}h\mathrm{\times }R\mathrm{\times }\frac{1}{n^2}$

If so, then the energy lost by an electron in the second equation above would be

$E_m{\text{-E}}_n\mathrm{=}\mathrm{-}h\mathrm{\times }R\mathrm{\times }\left(\frac{1}{m^2}\mathrm{-}\frac{1}{n^2}\right)\mathrm{=}\mathit{h\nu }$

This equation is the same as the Rydberg equation found experimentally. Therefore, we can conclude that, in a hydrogen atom, the energy of an electron can only be certain values given by an integer index n and equal to

$E_n\mathrm{=}\mathrm{-}h\mathrm{\times }R\mathrm{\times }\frac{1}{n^2}$

This means that the electron in a hydrogen atom can only exist in certain states with certain energies. These states must therefore determine the motion of the electron in the atom. Interestingly, this state of the electron is characterized by an integer, n , which we will now call a “quantum number” since it completely determines the quantized energy of the electron.

This discussion has only been about the hydrogen atom. These results also apply generally to all atoms, since all atoms display only specific frequencies which they emit or absorb. Since only certain frequencies can be emitted by each atom, only certain energy losses are possible, and only certain energy levels are possible in each atom. However, the equation above applies only to the energy of a hydrogen atom, since the Rydberg equation only describes the experimental spectrum of a hydrogen atom.

Review and discussion questions

1. The photoelectric effect demonstrates that radiation energy is quantized into “packets” or photons. Explain how and why this observation is of significance in understanding the structure of atoms.
2. Explain how we can know that higher frequency light contains higher energy photons.

By John S. Hutchinson, Rice University, 2011