all about: splittings

So the electron in the hydrogen atom is just a particle in a spherically-symmetric 1/r potential… you’ve got a ladder of energy eigenvalues indexed by a quantum number n. The nth eigenvalue has degeneracy n2, but that’s cool; picking an axis z, the total angular momentum operator L and the z-axis angular momentum operator Lz give a complete set of commuting observables (together with H), so you get yer n, l, ml eigenstates.

And you think everything’s cool, everything’s ok. Wrong, because physics gets in the way of all this math fun. A number of physical effects (in various environments & regimes) break spherical symmetry and perturb our energy levels from their sweetly degenerate state. Here are some of them.

Normal Zeeman effect
An orbital with \(L_z\neq 0\) has a non-zero magnetic moment about the z-axis (spinning electron ⇒ little loop of current ⇒ magnetic dipole). This means that it interacts with the z-component of a magnetic field. The potential of this interaction is \(V=-\mu B_z\) where μ is the dipole moment. How to calculate this? \(L_z = m_e v r\) and \(\mu = I A = \frac{-e v}{2\pi r} \pi r^2 = -\frac{e}{2} v r\): comparing these gives \(\mu = -\frac{e}{2 m_e} L_z\). (The coefficient in front is the “Bohr magneton”, within a factor of \(\hbar\).)

Since the spacing between Lz-values is \(\hbar\), the spacing between the split energy levels will be \((e\hbar B_z)/(2 m_e)\), which is the Lamor frequency times \(\hbar\).

Also interesting: Due to some symmetry magic, the eigenstates remain the same; only the eigenvalues change. Huh.

Anomalous Zeeman effect
Thing is, we’ve been talking about orbital angular momentum this whole time. There’s also the electron’s intrinsic “spin” angular momentum. This has can have magnitude \(\pm\hbar/2\) in the z direction. WEIRDLY enough, however, when computing a magnetic moment from this, you have to multiply by a “g-factor” of two. This shouldn’t surprise you, though, because spin was really weird in the first place.

The end result of all of this: Each state you get from normal Zeeman splitting splits \(\pm (e\hbar B_z)/(2 m_e)\), which was the spacing between these states in the first place. We get two extra energy levels.

Spin–orbit interaction / Fine structure
An orbiting electron sees a magnetic field produced by the movement of the nucleus. This magnetic field interacts with the spin dipole: when the field is parallel to the dipole (that is, anti-parallel to the spin), energy is minimized. Two possible values of spin ⇒ a new splitting. This produces a feature of atomic spectra known as fine structure doublets.

Stark effect
Just like an external magnetic field breaks symmetry and causes splitting in the Zeeman effect, an external electric field will cause a form of splitting known as the Stark effect. The reason why I put this in a second, subsidiary position is because it is way more complicated: our Hamiltonian now has a 1/r+z potential, which is going to completely screw up the old energy eigenstates. So you do perturbation theory, and get first/second-order corrections in the small-field limit. That’s quantum mechanics for you, I guess.

Hyperfine structure
This is real small. It includes things like electron-dipole/nuclear-dipole and electric-field/nuclear-quadrupole interactions.

Oh, and when it comes to computing spectra from these: have I mentioned selection rules? No? Oh, well those are pretty important too better look them up.