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.

decompo.PNG

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