PX262 - F3 - 3D harmonic oscillator

the potential energy operator: $\hat{V} (\vec{r}) = \frac{1}{2} k_{x} x^{2} + \frac{1}{2} k_{y} y^{2} + \frac{1}{2} k_{z} z^{2} = {\hat{V}}_{x} (x) + {\hat{V}}_{y} (y) + {\hat{V}}_{z} (z)$ where, $k_{x}, k_{y}, k_{z}$ are effective spring constants in the respective directions
solving the time-independent schrödinger equation by separation of variables: $- \frac{ℏ^{2}}{2 m} \nabla^{2} (X Y Z) + V (\vec{r}) X Y Z = E X Y Z$
dividing by $X Y Z$ and grouping terms with like coordinates: $[\frac{- ℏ^{2}}{2 m} \frac{1}{X} \frac{\partial^{2} X}{\partial x^{2}} + V_{x}] + [\frac{- ℏ^{2}}{2 m} \frac{1}{Y} \frac{\partial^{2} Y}{\partial y^{2}} + V_{y}] + [\frac{- ℏ^{2}}{2 m} \frac{1}{Z} \frac{\partial^{2} Z}{\partial z^{2}} + V_{z}] = E$
this can be split into three 1D harmonic oscillator equations, with three quantum numbers, $n_{x}, n_{y}, n_{z}$
the energy eigenstates are: $E_{n_{x} n_{y} n_{z}} = (n_{x} + \frac{1}{2}) ℏ ω_{x} + (n_{y} + \frac{1}{2}) ℏ ω_{y} + (n_{z} + \frac{1}{2}) ℏ ω_{z}$ where, $ω_{i}$ is related to the spring constant, $k_{i}$
the wavefunctions are: $ϕ (\vec{r}) = H_{n_{x}} (x^{'}) H_{n_{y}} (y^{'}) H_{n_{z}} (z^{'}) \exp (- \frac{1}{2} (x^{' 2} + y^{' 2} + z^{' 2}))$ where, $i^{'} = (\frac{M ω_{i}}{ℏ})^{\frac{1}{2}}$ for $i = {x, y, z}$
it can be observed that non-different $k_{i}$ values will lead to degenerate systems