Hydrogen Atom in a Weak Magnetic Field

One way to break the spherical symmetry is to apply an external B field. Lets assume that the field is weak enough that the energy shifts due to it are smaller than the fine structure corrections. Our Hamiltonian can now be written as \bgroup\color{black}$H=H_0+(H_1+H_2)+H_3$\egroup, where \bgroup\color{black}$H_0={p^2\over{2\mu}}-{Ze^2\over r}$\egroup is the normal Hydrogen problem, \bgroup\color{black}$H_1+H_2$\egroup is the fine structure correction, and

\begin{displaymath}\bgroup\color{black}H_3={e\vec B\over{2mc}}\cdot(\vec L +2\vec S)={eB\over{2mc}}(L_z +2S_z)\egroup\end{displaymath}

is the term due to the weak magnetic field.

We now run into a problem because \bgroup\color{black}$H_1+H_2$\egroup picks eigenstates of \bgroup\color{black}$J^2$\egroup and \bgroup\color{black}$J_z$\egroup while \bgroup\color{black}$H_3$\egroup picks eigenstates of \bgroup\color{black}$L_z$\egroup and \bgroup\color{black}$S_z$\egroup. In the weak field limit, we can do perturbation theory using the states of definite \bgroup\color{black}$j$\egroup. A direct calculation of the Anomalous Zeeman Effect gives the energy shifts in a weak B field.

\bgroup\color{black}$\Delta E=\left<\psi_{n\ell jm_j}\left\vert{eB\over{2mc}}(L_...
={e\hbar B\over {2mc}}m_j \left(1\pm {1\over{2\ell +1}}\right)$\egroup
This is the correction, due to a weak magnetic field, which we should add to the fine structure energies.

\begin{displaymath}\bgroup\color{black}E_{njm_j\ell s}=-{1\over 2}\alpha^2mc^2\l...
...^3}\left[{1\over j+{1\over 2}}-{3\over 4n}\right]\right)\egroup\end{displaymath}

Thus, in a weak field, the the degeneracy is completely broken for the states $\psi_{njm_j\ell s}$. All the states can be detected spectroscopically.


The factor \bgroup\color{black}$\left(1\pm {1\over{2\ell +1}}\right)$\egroup is known as the Lande $g$ Factorbecause the state splits as if it had this gyromagnetic ratio. We know that it is in fact a combination of the orbital and spin g factors in a state of definite \bgroup\color{black}$j$\egroup.

In the strong field limit we could use states of definite \bgroup\color{black}$m_\ell$\egroup and \bgroup\color{black}$m_s$\egroup and calculate the effects of the fine structure, \bgroup\color{black}$H_1+H_2$\egroup, as a correction. We will not calculate this here. If the field is very strong, we can neglect the fine structure entirely. Then the calculation is easy.

\begin{displaymath}\bgroup\color{black}E=E^0_n+{eB\hbar\over{2mc}}(m_\ell +2m_s)\egroup\end{displaymath}

In this limit, the field has partially removed the degeneracy in \bgroup\color{black}$m_\ell$\egroup and \bgroup\color{black}$m_s$\egroup, but not \bgroup\color{black}$\ell$\egroup. For example, the energies of all these \bgroup\color{black}$n=3$\egroup states are the same.

\begin{displaymath}\bgroup\color{black}\matrix{ \ell=2 & m_\ell=0 & m_s={1\over ...
...s={1\over 2} \cr
\ell=2 & m_\ell=2 & m_s=-{1\over 2} } \egroup\end{displaymath}

Jim Branson 2013-04-22