Question

The non-relativistic Hamiltonian for a single electron atom is
$H_0 = \frac{p^2}{2m} - V(r)$
where $V(r)$ is the Coulomb potential and $m$ is the mass of the electron.Considering the spin-orbit interaction term:
$H' = \frac{1}{2m^2c^2} \frac{1}{r} \frac{dV}{dr} \overrightarrow{L} \cdot \overrightarrow{S}$
added to $H_0$, which of the following statements is/are true?

• A. $H'$ commutes with $L^2$.
• B. $H'$ commutes with $L_z$ and $S_z$.
• C. For a given value of principal quantum number $n$ and orbital angular momentum quantum number $l$, there are $2(2l + 1)$ degenerate eigenstates of $H_0$.
• D. $H_0$, $L^2$, $S^2$, $L_z$, and $S_z$ have a set of simultaneous eigenstates.

Answer 1

Answer: (A), (C), (D)

$H'$ commutes with $L^2$.

Answer 2

The correct Answers are (A):$H'$ commutes with $L^2$.,(C):For a given value of principal quantum number $n$ and orbital angular momentum quantum number $l$, there are $2(2l + 1)$ degenerate eigenstates of $H_0$. ,(D): $H_0$, $L^2$, $S^2$, $L_z$, and $S_z$ have a set of simultaneous eigenstates,