Fine Structure Corrections

Beyond the Bohr model

The Bohr / Schrödinger hydrogen levels $E_n = -13.6\,\text{eV}/n^2$ ignore two effects that the relativistic treatment includes. Together they constitute the fine structure, and both are computed with perturbation theory because each is small compared to the gross level spacing.

1. The relativistic kinetic correction

The exact relativistic kinetic energy $\sqrt{p^2 c^2 + m^2 c^4} - mc^2$ expands as $\tfrac{p^2}{2m} - \tfrac{p^4}{8m^3c^2} + \cdots$. The leading correction beyond the Schrödinger kinetic term is

2. The spin–orbit coupling

In the electron's rest frame the nucleus orbits it, producing a magnetic field that couples to the electron's spin. After the Thomas factor of $1/2$, this gives

The $\vec{S}\cdot\vec{L}$ structure is exactly why total angular momentum $j$ (not $l$ and $s$ separately) labels the good states, and why the Landé factor of the previous lesson appears.

The size of the effect

Both corrections turn out to be of the same order and combine into the remarkably compact fine-structure energy

which depends on $n$ and $j$ but not on $l$ — a non-trivial cancellation between the two effects.

The crucial scaling is in the prefactor. Since $E_n^{(0)} \sim \alpha^2 mc^2$ (the Bohr energy in terms of the fine-structure constant $\alpha = e^2/4\pi\varepsilon_0\hbar c \approx 1/137$), we have

Fine structure is therefore about five parts in $10^5$ of the gross structure — small enough that first-order perturbation theory is excellent, but large enough to be measured precisely in spectra. This $\alpha^2$ suppression is why the Bohr model works so well despite ignoring relativity and spin.

Try it

Using $\alpha = 1/137.036$, compute the order-of-magnitude suppression factor $\alpha^2$ that separates fine structure from the gross structure, and return it as a number.

Sign in on the full site to ask questions and join the discussion.