Franz Schwabl Quantum Mechanics 4th Edit · Problem 12.2

Several-Electron Atoms

Physics — Quantum Mechanics · spin-orbit-harmonic-oscillator

Problem

Study the influence of the spin--orbit interaction

H_2 = \frac{1}{2m^2c^2}\boldsymbol{S} \cdot \boldsymbol{L}\,\frac{1}{r}\frac{dV(r)}{dr}

on the energy spectrum

\left(E_{n,l} = \hbar\omega\left(l + 2n + \frac{3}{2}\right)

, compare problem 17.1

\right)

of a three-dimensional isotropic harmonic oscillator. Discuss the degeneracy of the energy levels without and with the spin--orbit interaction. Note:

\psi_{nlm}(\boldsymbol{x}) = R_{nl}(r)\,Y_{lm}(\vartheta,\varphi)