Past exam of the atomic and laser physics course of the University of Oxford / 2021 / 1 / d

Neglect hyperfine structure, spin-orbit coupling and the diamagnetic term, and set $b={\mu }_{B}B$, $\epsilon =\alpha a_0^2$. The Zeeman energy is $b(L_{\hat{B}}+2S_{\hat{B}})$. The electron spin is free, so every spatial result below is duplicated with an added $\pm b$.

For a magnetic field parallel to $z$, the old magnetic quantum number $m$ remains good and the first-order energy levels areThe two $s$ orbitals have zero orbital angular momentum, so only their electron spin contributes at this order.

For a magnetic field parallel to $x$, the expectation of $L_x$ in each individual $p_m$ orbital vanishes. One must still check the degenerate $m=\pm 1$ subspace: its projected $L_x$ is zero as well. Thus there is no first-order orbital splitting in a sufficiently weak transverse magnetic field. Spin still contributes $\pm b$.

The transverse orbital Zeeman mixing can be seen without assuming its size. Put $E_0=-\mathrm{Ry}/4+3\epsilon$, $E_t=-\mathrm{Ry}/4+21\epsilon$, and $\Delta =E_t-E_0=18\epsilon$. In the real $p_x,p_y,p_z$ orbital basis, $p_x$ is unaffected while $p_y,p_z$ have off-diagonal matrix elements of modulus $b$. Diagonalizing this two-dimensional Hamiltonian gives the spatial energy eigenvaluesWhen $b\ll \Delta$, their shifts relative to the earlier levels are $0,+b^2/\Delta ,-b^2/\Delta$, with $\pm b$ added for electron spin. The displayed square-root expression also handles $b$ comparable to $\Delta$, provided coupling to other principal shells stays negligible. The $2s$ orbital retains zero orbital correction.

These quadratic orbital estimates keep the enhanced contribution from mixing the split $2p$ levels. Minimal magnetic minimal coupling also produces $e^2{B}^2{r}_{\perp }^2/(8m_e)$; its expectation value is positive and must be added for a full order-$B^2$ calculation. For example it gives $e^2{B}^2{a}_0^2/(4m_e)$ for $1s$. It is parametrically smaller than the $2p$ mixing term when $\epsilon \ll \mathrm{Ry}$.