# Degeneracy of the Quantum Linear Oscillator

mavanhel
So, today while doing my homework for statistical mechanics I was reading about the quantum linear oscillator in the textbook, "Classical and Statistical Thermodynamics" by Ashley H. Carter. In it, after discussing the quantized energy it says:

"Note that the energies are equally spaced and that the ground state has a 'zero-point' energy equal to $$\frac{hv}{2}$$. The states are nondegenerate in that $$g_{j} = 1$$ for all $$j$$."

The book gives no further clarification on this point as, but I was wondering why the degeneracy for this problem is one for all states.

This problem occurs at the beginning of chapter 15, "The Heat Capacity of a Diatomic Gas" and makes the following considerations:
• consider an assemply of N one-dimensional harmonic oscillators.
• The oscillators are loosely coupled (ie. small energy exchange between them).
• The oscillators are free to vibrate in one dimension freely.

This does not tell us anything about the degeneracy, so why is this system nondegenerate?