I'm trying to determine the degeneracy of states given by $g(\epsilon)=g_{0} \epsilon$ for a system that is trapped in a quite specific potential.
In two dimensions, the particle has a potential as in the infinite square well, while on the third dimension, the potential behaves as if a harmonic oscillator.
The energy for a single particle is given by
$$\epsilon = \frac{h^{2}}{8mL^{2}}(n_{x}^{2} + n_{y}^{2}) + hf \biggl(n_{z} – \frac{1}{2}\biggr)$$
I'm having some difficulty with this problem because of the broken symmetry. Is there any analytic way to determine the degeneracy as a function of energy?
Best Answer
This seems to scream out for a Separation of Variables approach. In addition to those links you can find it in any book on math methods in physics or any reasonably advanced book on differential equations.
The short--short version is you will write you solution in parts that depend only on the independent bits
$$ \Psi(x,y,z) = W(x,y) Z(z) $$
and after applying the partial derivatives you will find that you have two much simpler equations to work with.