23 Fine Structure in Hydrogen
In this section, we will calculate the fine structure corrections to the Hydrogen spectrum. Some of
the degeneracy will be broken. Since the Hydrogen problem still hasspherical symmetry, states of
definite total angular momentum will be the energy eigenstates.
We will break the spherical symmetry by applying a weak magnetic field, further breaking the
degeneracy of the energy eigenstates. The effect of a weak magnetic field is known as the anomalous
Zeeman effect, because it was hard to understand at the time it wasfirst measured. It will not be
anomalous for us.
We will use many of the tools of the last three sections to make our calculations. Nevertheless, a
few of the correction terms we use will not be fully derived here.
This material is covered inGasiorowicz Chapter 17,inCohen-Tannoudji et al. Chapter
XII,and inGriffiths 6.3 and 6.4.
23.1 Hydrogen Fine Structure
The basic hydrogen problem we have solved has the following Hamiltonian.
H 0 =
p^2
2 μ−
Ze^2
rTo this simple Coulomb problem, we will addseveral corrections:
- The relativistic correction to the electron’s kinetic energy.
- The Spin-Orbit correction.
- The “Darwin Term” correction to s states from Dirac eq.
- The ((anomalouus) Zeeman) effect of an external magnetic field.
Correction (1) comes fromrelativity. The electron’s velocity in hydrogen is of orderαc. It is not
very relativistic but a small correction is in order. Bycalculating(see section 23.4.1)the next
order relativistic correction to the kinetic energy we find the additional term in the Hamiltonian
H 1 =−
1
8
p^4 e
m^3 c^2.
Our energy eigenstates are not eigenfunctions of this operator so we will have totreat it as a
perturbation.
We canestimate the sizeof this correction compared to the Hydrogen binding energy by taking
the ratio to the Hydrogen kinetic energy. (Remember that, in the hydrogen ground state,
〈
p^2
2 m〉
=
−E=^12 α^2 mc^2 .)
p^4
8 m^3 c^2
÷
p^2
2 m=
p^2
4 m^2 c^2=
(p^2 / 2 m)
2 mc^2=
1
4
α^2