Fine structure and spin-orbit coupling

The simple hydrogen energy formula treats the electron as a point charge orbiting a fixed nucleus, and predicts that energy depends only on the principal quantum number n — every state with the same n, regardless of its orbital shape or spin, comes out with exactly the same energy. That is an approximation. Switch to the electron's own point of view and the picture changes: from where the electron sits, it is the positively charged nucleus that appears to orbit around it, and a moving charge creates a magnetic field. The electron's own spin has a magnetic moment, and that magnetic moment interacts with the field its "orbiting" nucleus produces — a purely internal effect that needs no external field at all.

This spin-orbit interaction shifts each energy level up or down slightly depending on how the electron's spin angular momentum and orbital angular momentum combine into a total angular momentum, described by the quantum number j. For a given orbital quantum number l, j can only take the values l + 1/2 or l − 1/2, and each gets a slightly different energy. This is why a spectral line that the simple model predicts as a single frequency is actually a close pair when examined precisely — the "fine structure" that gives the effect its name.

One state is conspicuously exempt: when l = 0 (an s-state), the electron has no orbital angular momentum in the first place, so there is no orbital magnetic field for its spin to couple to, and no spin-orbit splitting occurs. Every p, d and f state, by contrast, splits into two closely spaced sub-levels.