This time we will project the deformed orbit into spherical orbit. The wavefunctions are stated in here again.

$\displaystyle \Psi^D_{N n_z m}(z, \rho, \phi) = |Nn_z m\rangle_D \\ = \sqrt{\frac{1}{\alpha_z \alpha^2}}\sqrt{\frac{ n_\rho !}{2^n_z n_z! (m + n_\rho)!\sqrt{\pi^3}}} H_{n_z}\left(\frac{z}{\alpha_z}\right) \\ \exp\left(- \frac{1}{2}\left(\frac{z^2}{\alpha_z^2}+\frac{\rho^2}{\alpha^2}\right)\right) \left(\frac{\rho}{\alpha}\right)^{m} L_{n_\rho}^{m} \left(\frac{\rho^2}{\alpha^2}\right) \exp(i m \phi)$

$\displaystyle \Psi^S_{Nlm}(r, \theta, \phi) = |N l m_l\rangle_S\\ =\sqrt{ \frac{1}{\sqrt{\pi}\alpha^{2l+3}} \frac{(\frac{N-l}{2})! (\frac{N+l}{2})! 2^{N+l+2}}{(N+l+1)!}} r^l \exp\left(-\frac{r^2}{2\alpha^2}\right) L_{k}^{l+\frac{1}{2}}\left( \frac{r^2}{\alpha^2} \right) Y_{lm}(\theta, \phi)$

Since both functions span the entire space and are basis, thus, we can related them as

$\displaystyle|Nn_z m \rangle_D = \sum_{N' l' m'} C_{N n_z m}^{N' l' m'} |N' l' m' \rangle_S$

where

$latex C_{N n_z m}^{N’ l’ m’} = \langle (N’l’m’)_S|(N n_z m)_D \rangle$

First thing we notice is that $m' = m$ . Because the $phi$ components are the same in both wave function. i.e. $\int \exp(- i m' \phi) \exp(i m \phi) d\phi = 0$ if $m' \neq m$. We can omit the $m, m'$, so that $C_{N n_z}^{N' l'}$

Second thing is the parity must be the same, thus when $N$ is even (or odd), $N'$ must be even (or odd).

For non-deformed $\delta = 0$, $N' = N$,, here are some results

$\displaystyle|0 0 0 \rangle_D = |0 0 0\rangle_S$

$\displaystyle|1 1 0 \rangle_D = |1 1 0 \rangle_S$

$\displaystyle|1 0 1 \rangle_D = |1 1 1 \rangle_S$

$\displaystyle|2 2 0 \rangle_D = \sqrt{\frac{2}{3}} |2 2 0 \rangle_S - \sqrt{\frac{1}{3}} |2 0 0 \rangle_S$

$\displaystyle|2 1 1 \rangle_D = |2 2 1 \rangle_S$

$\displaystyle|2 0 2 \rangle_D = |2 2 2 \rangle_S$

$\displaystyle|2 0 0 \rangle_D = \sqrt{\frac{1}{3}} |2 2 0 \rangle_S + \sqrt{\frac{2}{3}} |2 0 0 \rangle_S$

$\displaystyle|3 3 0 \rangle_D = \sqrt{\frac{2}{5}} |3 3 0 \rangle_S - \sqrt{\frac{3}{5}} |3 1 0 \rangle_S$

$\displaystyle|3 2 1 \rangle_D = -\sqrt{\frac{4}{5}} |3 3 1 \rangle_S + \sqrt{\frac{1}{5}} |3 1 1 \rangle_S$

$\displaystyle|3 1 2 \rangle_D = |3 3 2 \rangle_S$

$\displaystyle|3 0 3 \rangle_D = |3 3 3 \rangle_S$

$\displaystyle|3 1 0 \rangle_D = \sqrt{\frac{3}{5}} |3 3 0 \rangle_S + \sqrt{\frac{2}{5}} |3 1 0 \rangle_S$

$\displaystyle|3 0 1 \rangle_D = -\sqrt{\frac{2}{5}} |3 3 1 \rangle_S - \sqrt{\frac{4}{5}} |3 1 1 \rangle_S$

For $\delta = 0.3$

$\displaystyle|0 0 0 \rangle_D = 0.995|0 0 0 \rangle_S + 0.099|2 2 0 \rangle_S - 0.015|2 0 0 \rangle_S + 0.009|4 4 0 \rangle_S + ...$

$\displaystyle|1 1 0 \rangle_D = 0.987|1 1 0 \rangle_S + 0.132|3 1 0 \rangle_S - 0.014|5 3 0 \rangle_S + 0.014|5 5 0 \rangle_S + ...$

$\displaystyle|1 0 1 \rangle_D = 0.994|1 1 1 \rangle_S + 0.108|3 3 1 \rangle_S + 0.017|3 1 1 \rangle_S + 0.011|5 5 1 \rangle_S + ...$

$\displaystyle|2 2 0 \rangle_D = 0.790|2 2 0 \rangle_S - 0.564|2 0 0 \rangle_S - 0.153|4 2 0 \rangle_S + 0.139|4 4 0 \rangle_S + ...$

$\displaystyle|2 1 1 \rangle_D = 0.986|2 2 1 \rangle_S + 0.158|4 4 1 \rangle_S - 0.052|4 2 1 \rangle_S - 0.012|6 4 1 \rangle_S + ...$

For $\delta = 0.6$

$\displaystyle|0 0 0 \rangle_D = 0.958|0 0 0 \rangle_S + 0.258|2 2 0 \rangle_S - 0.084|2 0 0 \rangle_S + 0.061|4 4 0 \rangle_S + ...$

$\displaystyle|1 1 0 \rangle_D = 0.885|1 1 0 \rangle_S + 0.319|3 3 0 \rangle_S - 0.274|3 1 0 \rangle_S - 0.119|5 3 0 \rangle_S + ...$

$\displaystyle|1 0 1 \rangle_D = 0.955|1 1 1 \rangle_S + 0.281|3 3 1 \rangle_S + 0.078|5 5 1 \rangle_S - 0.043|5 3 1 \rangle_S + ...$

$\displaystyle|2 2 0 \rangle_D = 0.598|2 2 0 \rangle_S - 0.45|2 0 0 \rangle_S - 0.379|4 2 0 \rangle_S + 0.299|4 4 0 \rangle_S \\ - 0.259|0 0 0 \rangle_S + 0.198|4 0 0 \rangle_S + 0.173|6 2 0 \rangle_S - 0.168|6 4 0 \rangle_S + ...$

$\displaystyle|2 1 1 \rangle_D = 0.882|2 2 1 \rangle_S + 0.381|4 4 1 \rangle_S - 0.191|4 2 1 \rangle_S - 0.129|6 6 1 \rangle_S \\ - 0.114|6 4 1 \rangle_S + ...$

We can see, more deform, more higher angular momentum states are involved.

Also, for a pure state when non-deform, the mixing is still small.