An Introduction to Thermal Physics · Problem A.18

A three-dimensional isotropic oscillator is really three independent one-dimensional oscillators stacked together—each direction contributes its own quantum number. The key insight is that the total energy depends only on the sum of the three quantum numbers, not on how that sum is distributed among $n_x$, $n_y$, and $n_z$. So degeneracy counts the number of ways to partition a given total quantum number into three non-negative integer parts—a combinatorics problem, not a physics one.

A three-dimensional isotropic oscillator is really three independent one-dimensional oscillators stacked together—each direction contributes its own quantum number. The key insight is that the total energy depends only on the sum of the three quantum numbers, not on how that sum is distributed among $n_x$, $n_y$, and $n_z$. So degeneracy counts the number of ways to partition a given total quantum number into three non-negative integer parts—a combinatorics problem, not a physics one.