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Generating Functions. Part 2. Degeneration of Energy Levels in Central Field.

Seemingly intimidating task of calculating degeneration of energy levels for quantum particle in central field can be solved easily with generating function.

Degeneration of Energy Levels in Central Field

Problem. For electron in central field in R^3 find degeneration of its energy levels. 

Note that the actual shape of the potential field doesn’t matter as long as it is spherically symmetric (and not pathological) . Thus the answer equally applies to harmonic oscillator and hydrogen atom [1].

Hamiltonian in 3D has basis of eigenfunctions formed by three operators corresponding to impulses along each of the coordinates. Hence  the energy is uniquely defined by sum of the eigenvalues for these operators (we’re obviously skipping important technical details that justify this conclusion):

(1) n=n_x+n_y+n_z

The degeneration of energy level for particular value of n is the number of ways it can be combined from three non-negative integers. The number of such combinations comes from generating function:

(2) F(x)=(1+x+x^2+ x^3+ . . . )^3=\frac{1}{(1-x)^3}

Using general formula for expansion:

(3) (1-x)^{-n}  =\sum_{s=0}^{\infty}C_{n-1+s}^{n-1} x^s

we obtain:

(4) F(x)=\sum_{s=0}^{\infty}C_{2+s}^{2} x^s

For n=0 we get coefficient 1, for n=1 we get 3, …, for n=10 the degeneration is 66, etc. A shorter expression gives the textbook formula:

(5)  A_n=\frac{1}{2}(1+n)(2+n)

Paper [2] fills in some of the details we skipped here and also discusses another application of generating function to counting in elementary quantum mechanics. The emphasis here is on elementary, of course, as the discussed problems do not go beyond typical textbook. Back in mid-80s I used this solution for central field on the final on quantum mechanics at MIPT. And that prompted the professor to give me another problem – counting “lucky tickets”, which we will discuss in the next post.


[1] Bes D.R. “Quantum Mechanics. A Modern and Concise Introductory Course”, Springer 2007,  255 pages.
[2] Li Han “Generating Function in Quantum Mechanics: An Application to Counting Problems“,  arXiv:0711.2854 [physics.gen-ph].