M2L7j.txt

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The fact that we have spin and we have spin-orbit interaction
means that the spin-orbit interaction L.S is not
diagonal in L or S, so what we have to do is we
have to introduce now the total angular momentum J.
And since L plus S square equals J square,
we can write the product LS as 1/2 J square
minus L square minus S square.
So therefore the LS interaction is diagonal in the J basis.
So if you couple L and S to J, a given state of the operator J
has a definite value for LS.
Let's put in things together.
So at the beginning of this class,
we started out with a hydrogen atom--
where let me just focus on n equals 2-- where we had n
equals 2, all n equals 2 states have the same energy.
That's what comes out of the non-relativistic Schrodinger
equation with a Coulomb potential,
and that means we had a degeneracy for the 2P 1/2,
for the 2P 3/2, and the 2S 1/2 state.

If you now put in spin-orbit interaction, well,
nothing of course happens to the S 1/2 state
because it has no orbital angular momentum.
And then in the P 3/2 state, l equals
1, one unit of orbital angular momentum,
and the spins are aligned.
In the P 1/2 state, they're anti-aligned.
So therefore, the two terms have opposite shifts.
So the P 3/2 state is shifted up and the P 1/2 state
is shifted down.
But then in the second step, we want
to add in now the two other contributions
to the fine structure, the kinetic and the Darwin term,
and then something happens which is really remarkable.
We have split the three levels.
We have lifted the degeneracy between the three levels.
But if you now add the kinetic and the Darwin term,
it turns out that the S 1/2 and the P 1/2 states
become exactly degenerate again, so there is still
a degeneracy in the spectrum.

If we would use a non-relativistic approach
and derive the Darwin term, the spin-orbit term,
and the relativistic correction separately,
there would be no reason.
It would just look like a freak accident in nature
that those two levels come out equal.
However, it's not a freak accident,
it's a symmetry of the Dirac equation.
So all those corrections have a deep connection
in relativistic physics, and relativistic physics
preserves the degeneracy in j.
So let me write that down.
So the fine structure does not lift the degeneracy
between S 1/2 and P 1/2.
It turns out that when we use the Dirac equation,
we can get an exact expression for the fine structure
n to the four, n is the principle quantum number,
and then the fine structure only depends on j and not on l
and s separately.
So that tells us that eventually,
the spin of the electron and the fine structure really
have deep origins in the relativistic nature
of the underlying physics.

Any questions about fine structure?