Degeneracy of Hydrogen Atom

The energy levels of the hydrogen atom are only dependent on the principal quantum number n and not on other quantum numbers l and m. Thus, by knowing the principle quantum number, the energy levels of the hydrogen atom can be determined. For any value of principle quantum number n, the possible values of l are, l = 0, 1, 2, 3…n − 1. For each l, there are (2l + 1) possible values of m.
This means that many quantum states have the same energy, and therefore, the system is degenerate. For ground state, n = 1, l = 0, and m = 0. For n = 2, energy state, l has two values, l = 0, 1 and for each l, there are corresponding m values. For l = 0, m = 0, and for l = 1, there are three values of m, as m=−1, 0, +1. Therefore, the total degeneracy of the energy level E_n is:

This property of energy dependence mainly on the principle quantum number is characteristic of the hydrogen atom only and is due to the nature of the coulombic potential. In other atoms, energy is dependent not only on “n” but also on the “l” quantum number, since their potential function deviates from the coulombic potential due to screening effect of other electrons.
Historically, the principle quantum number “n” is called a shell and is denoted as s. Each shell contains n subshells corresponding to the quantum number “l.” Each subshell has (2l + 1) states associated with the quantum number “m.” These energy states of the hydrogen atom are depicted in Figure 1.

FIGURE 1: Energy states distribution of hydrogen atom, shell and subshells are depicted.