The Nilsson orbital can be decomposed into series of orbitals of  3D-harmonic oscillator, such that

\displaystyle [Nn_z\lambda]K = \sum_{N'lj} C_{N'lj}^{N n_z \lambda} |N'ljK\rangle

with eigen energy \epsilon_{Nn_z\lambda K} (\beta) and

n_z + \lambda = l

n_z + K = j


Since the Nilsson orbital is normalized

\displaystyle \sum_{N'lj} \left(C_{N'lj}^{N n_z \lambda}\right)^2 = 1

Since the number of orbital for fixed l,j is 2j+1, thus using an inverse transformation from spherical orbital to Nilsson orbital, we have,

\displaystyle \sum_{N n_z \lambda} \left(C_{N'lj}^{N n_z \lambda}\right)^2 = 2j+1


I cannot prove it, but

\displaystyle \sum_{N n_z \lambda} \epsilon_{N n_z \lambda K}(\beta) \left(C_{N'lj}^{N n_z \lambda}\right)^2 = \epsilon_{N n_z \lambda K} (0)

Thus, the single-particle energy fro Nilsson orbital is as same as the spherical orbital !!!

 

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