Degenerate Perturbation Theory for Spin-Orbit Coupling

Including spin, the level $E_n$ of hydrogen belongs to the states

$$\vert n l s m_l m_s\rangle,\quad s=1/2,\quad m_s=\pm 1/2,$$ (3.20)

which are eigenstates of $\hat{L}^2$, $\hat{S}^2$, $\hat{L}_z$, and $\hat{S}_z$ (`uncoupled representation'). With $\hat{\bf L}$ and $\hat{\bf S}$ adding up to the total angular momentum $\hat{\bf J} = \hat{\bf L} + \hat{\bf S}$, an alternative basis is the `coupled representation'

$$\vert n l s j m \rangle,\quad j=l+s, l+s-1, ..., \vert l-s\vert,\quad m=m_l+m_s.$$ (3.21)

of eigenfunctions of $\hat{J}^2$, $\hat{L}^2$, $\hat{S}^2$, and $\hat{J}_z$. Here, $s=1/2$ is the total electron spin which of course is fixed and gives the two possibilities $j=l+1/2$ and $j=l-1/2$ for $l\ge 1$ and $j=1/2$ for $l=0$ ( $l$ runs from 0 to $n-1$).

The perturbation $\hat{H}_{\rm SO}$, Eq. (II.3.12), can be diagonalised in the $\vert n l s j m \rangle$ basis, using

$$\hat{\bf S}\hat{\bf L} = \frac{1}{2}\left( \hat{\bf J}^2-\hat{\bf L}^2 - \hat{\bf S}^2 \right)$$ (3.22)

$$\leadsto \langle n l' s j' m'\vert \hat{\bf S}\hat{\bf L} \vert n l s j m \rangle = \frac{1}{2}\hbar^2 \left( j(j+1) - l(l+1) -s(s+1) \right)\delta_{jj'}\delta_{ll'}\delta_{mm'}.$$

For fixed $n$, $l$, and $m$, ( $s=1/2$ is fixed anyway and therefore a dummy index), the basis of degenerate states from the previous subsection therefore for $l\ge 1$ has two states, $\vert n l s j=l\pm 1/2 m \rangle$, and the two-by-two matrix $\underline{\underline{H}}$ is diagonal,

$$\underline{\underline{H}} \leftrightarrow \langle n l s j' m\vert... ...{2}\hbar^2\left( \begin{array}{cc} l & 0 \\ 0 & -(l+1) \\ \end{array}\right),$$ (3.23)

where $\left\langle \frac{1}{r^3} \right\rangle_{nl}$ indicates that this matrix elements has to be calculated with the radial parts of the wave functions $\langle {\bf r} \vert n l s j=l\pm 1/2 m \rangle$, with the result

$$\left\langle \frac{1}{r^3} \right\rangle_{nl} = \frac{Z^3}{a_0^3}\frac{2}{n^3l(l+1)(2l+1)},\quad l\ne 0.$$ (3.24)

The resulting energy shifts $E'_{\rm SO}$ corresponding to the two states with $j=l\pm 1/2$ are

$$E'_{\rm SO}=\frac{Z^4e^2\hbar^2}{2m^2c^2a_0^3 4\pi\varepsilon... ...+\frac{1}{2}\\ -(l+1), & j=l-\frac{1}{2}\\ \end{array}\right.$$ (3.25)