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 *n*^{th} eigenvalue has degeneracy *n*^{2}, but *that’s cool*; picking an axis *z*, the total angular momentum operator * L* and the

*z*-axis angular momentum operator

*L*give a complete set of commuting observables (together with

_{z}*H*), so you get yer

*n*,

*l*,

*m*eigenstates.

_{l}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

*L*-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 \(\hbar\)._{z}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.

__Spinâ€“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.