Quantized Energy Levels in Hydrogen Atoms

Author: John Hutchinson

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

The Hydrogen atom spectrum also tells us what these energy levels are. Recall that the frequencies of radiation emitted by Hydrogen atoms are given by the Rydberg equation. Each choice of the positive integers n and m predicts a single observed frequency in the hydrogen atom spectrum.

Each emitted frequency must correspond to an energy hν by Einstein’s equation. This photon energy must be the difference between two energy levels for a hydrogen electron, since that is the amount of energy released by the electron moving from one level to the other. If the energies of the two levels are Em and En, then we can write that

hν = Em - En

[4]

By comparing this to the Rydberg equation, each energy level must be given by the formula

En = (-h)R/n2

[5]

We can draw two conclusions. First, the electron in a hydrogen atom can exist only with certain energies, corresponding to motion in what we now call a state or an orbital. Second, the energy of a state can be characterized by an integer quantum number, n = 1, 2, 3, ... which determines its energy.

These conclusions are reinforced by similar observations of spectra produced by passing a current through other elements. Only specific frequencies are observed for each atom, although only the hydrogen frequencies obey the Rydberg formula.

We conclude that the energies of electrons in atoms are "quantized," that is, restricted to certain values. We now need to relate this quantization of energy to the existence of shells, as developed in a previous study.