Long time ago, I tried to tackle the Nilsson orbital by solving the Hamiltonian analytically. However, the Hamiltonian is without LS coupling. This times, I redo the calculation according to the reference B. E. Chi, Nuclear Phyiscs 83 (1966) 97-144.

The Hamiltonian is

using

The Hamiltonian becomes

Set , and

Set

and the perturbation is

The wavefunction for the spherical harmonic is

The diagonal elements are

where

The off diagonal elements are

( I will evaluate this integral in future )

The rest is diagonalization the Hamiltonian

Here is the calculation for the 2nd harmonic for

The component of each orbital can be directly taken from the eigenvalue. Here is the [521]1/2 state.

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