130_notes.dvi

A problem arises in the case ofdegenerate statesor nearly degenerate states. The energy denom-
inator in the last equation above is small and the series does not converge. To handle this case, we
need to rediagonalize the full Hamiltonian in the subspace of nearly degenerate states.

∑

i∈N

〈φn(j)|H|φ(ni)〉αi=Enαj.

This is just the standard eigenvalue problem for the full Hamiltonian inthesubspace of (nearly)
degenerate states.

We will use time independent perturbation theory is used to computefine structure and hyperfine
corrections to Hydrogen energies, as well as for many other calculations. Degenerate state pertur-
bation theory will be used for the Stark Effect and for hyperfine splitting in Hydrogen.

1.30 The Fine Structure of Hydrogen

We have solved the problem of a non-relativistic, spinless electron in acoulomb potential exactly.
Real Hydrogen atoms have several small corrections to this simplesolution. If we say that electron
spin is a relativistic effect, they can all be called relativistic corrections which are off orderα^2
compared to the Hydrogen energies we have calculated.

1. The relativistic correction to the electron’s kinetic energy.


2. The Spin-Orbit correction.


3. The “Darwin Term” correction to s states from Dirac equation.

Calculating these fine structure effects (See section 23) separately and summing them we find that
we get a nice cancellation yielding a simple formula.

Enlm=E(0)n +

E

(0)

n

2

2 mc^2

[

3 −

4 n

j+^12

]

The correction depends only on the total angular quantum numberand does not depend onℓso the
states of different total angular momentum split in energy but there is still a good deal of degeneracy.
It makes sense, for aproblem with spherical symmetry, that the states of definitetotal
angular momentum are the energy eigenstatesand that the result depend onj.

We also compute theZeeman effectin which an external magnetic field is applied to Hydrogen.
The external field is very important since it breaks the spherical symmetry and splits degenerate
states allowing us to understand Hydrogen through spectroscopy.

The correction due to aweak magnetic fieldis found to be

∆E=

〈

ψnℓjmj

∣

∣

∣

∣

eB

2 mc

(Lz+ 2Sz)

∣

∣

∣

∣ψnℓjmj

〉

=

e ̄hB

2 mc

mj

(

1 ±

1

2 ℓ+ 1

)

The factor

(

1 ± 2 ℓ^1 +1

)

is known as theLandegFactorbecause the state splits as if it had this

gyromagnetic ratio. We know that it is in fact a combination of the orbital and spin g factors in