| Principal Quantum Number n | Angular Momentum Magnitude l | Angular Momentum Projection m |
| 1 | 0 | 0 |
| 2 | 0 | 0 |
| 2 | 1 | -1, 0, 1 |
| 3 | 0 | 0 |
| 3 | 1 | -1, 0, 1 |
| 3 | 2 | -2, -1, 0, 1, 2 |
| 4 | 0 | 0 |
| 4 | 1 | -1, 0, 1 |
| 4 | 2 | -2, -1, 0, 1, 2 |
| 4 | 3 | -3, -2, -1, 0, 1, 2, 3 |
| n | 0 … n-1 | –l … l |
Here, each unique combination of n, l, and m corresponds to a different quantum state (e.g., n = 2, l = 1, m = -1 is one quantum state and n = 2, l = 1, and m = 0 is another). Each quantum state represents a three-dimensional wave function found by solving the Schroedinger equation for the hydrogen atom. However, as deduced by Bohr, the energy levels depend only on the principal quantum number n. The angular momentum doesn’t affect the energy of a state directly. That is to say, all the states with the same n have the same energy, even though they can have different values of m and l.
When multiple quantum states have the same energy (say, n = 2, l = 1, m = -1 and n = 2, l = 1, and m = 0), we say there is an energy degeneracy and that the quantum states are degenerate. Each value of n has a different degree of degeneracy. If you count the various combinations of quantum numbers in the table, you can see that n = 2 has four degenerate quantum states, n = 3 has nine degenerate quantum states, etc. The degree of degeneracy turns out to be equal to n2.
DEFINITION
Energy degeneracy describes the situation where distinct quantum states have identical energies.
This sums up the basics of the hydrogen atom. However, for a complete account of the hydrogen atom, and other atoms across the periodic table, we have to deal with one more important quantum feature—something we call spin.
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