Sakurai 5.24
Here we are told that the $n = 2$ state of the Hydrogen Atom is a $g = 8$-fold degenerate
state (when accounting for both spin, $s$, $m_s$, and the orbital angular momentum, $l$, $m_l$). We
are told that we can break (remove) the degeneracy by introducing the following perturbation
potential...
$V = \frac{A}{\hbar^2} (\vec{L} \cdot \vec{S}) + \frac{B}{\hbar} (L_z + 2S_z)$
...where the following quantities are defined as...
$\bullet \vec{L}$ = Orbital Angular Momentum Operator
$\bullet \vec{S}$ = Spin Angular Momentum Operator
$\bullet A$ = Constant
$\bullet B \propto B_z$ (Proportional Magnetic Field in z-Direction)
...we are then asked to...
a) Express the perturbed potential, $V$ in terms of $J^2$, $L^2$, $S^2$, $J_z$ and $S_z$, where as
usual, the total angular momentum, $\vec{J}$ is given by...
$\vec{J} = \vec{L} + \vec{S}$
