M2L7e.txt

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We can now proceed form electronic energies, which
were on the order of a Rydberg, or remember atomic units,
fine-structure constant squared times
rest mass of the electron.
So we've taken care of those energies.
And now we want to take care of smaller corrections.
And what we discuss next is the fine-structure
and the lamb shift.
So this is now the title of our next chapter, Fine-Structure
and Lamb Shift.
The fine-structure energies are alpha squared times smaller
than the Rydberg.
So they are on the order of alpha to the 4 mc squared.
And well, if I write that as alpha square Rydberg,
which is the same.
But this explains to you why alpha is called
the fine-structure constant.
People had learned about the hydrogen spectrum,
Bohr's model and all that.
And then they found finer corrections.
And those finer corrections were on a scale
alpha squared times smaller.
And the constant which appears here,
alpha, was called the fine-structure constant
because it described the structure
of the fine-structure corrections
and the fine-structure splittings.
Of course, if you want, you can make
the fine-structure constant already
appear for electronic energies, but this simply
reflects that I have used the rest mass of the electron
as a reference energy, which unless you
want to discuss fundamental units, doesn't make any sense.
Because c, the speed of light, does not
appear in the electronic structure as we
discussed earlier.
So therefore, the fine-structure constant
does enter the picture only when we now
discuss fine-structure corrections,
fine-structure splittings.
The Lamb shift, which we will then discuss,
is even higher order.
It's alpha to the fifth.

But just to tell you, the Lamb shift
is on the order of a gigahertz.
And you know that standard precision in all of your labs
is now megahertz or better, so the Lamb shift,
although it's alpha to the fifth, is a big effect
on this scale of how we understand and probe
the structure of atoms today.

So let's talk about the fine-structure.

You will actually find a complete discussion
of the fine-structure in several textbooks
on quantum mechanics-- Griffiths, Cohen-Tannoudji,
Gasiorowicz.
What I want to do here is to have a careful discussion
of the physical origin and provide
a clear understanding what is responsible
for the fine-structure.

So if you just want to get an accurate result,
there's only one thing you have to do, solve the Dirac equation.
The solution of the Dirac equation for the hydrogen atom
gives us the electronic structure,
the same as comes out of the Bohr model,
but now including the fine-structure.
So it's nice.
It just comes out.
You don't have to put in the spin of the electron by hand.
It comes out automatically.
But it also comes out as a result,
and you don't really understand what
is the origin of that, because you've just treated everything
fully relativistically.
So what I want to show you here is
I want to show you that we can distinguish
three contributions.
There are three physical effects which
all come in at the level alpha to the 4 mc square.
One is relativistic corrections to the kinetic energy.
The second one is spin orbit coupling.
And then there is a third one, which
goes by the name Darwin term.
And we can get this physical insight
and obtain those three terms separately
when we use what is called the Pauli
approximation to the Dirac equation.
If you do this so-called Pauli approximation
to the Dirac equation, you simply
have to expand the Dirac equation in powers of v over c.
And then you get the Schrodinger equation plus correction terms.
And what you obtain then is a Hamiltonian,
which is the rest energy-- which is
constant-- the non-relativistic kinetic energy, the Coulomb
energy.
But then we have three more terms,
and these are the three contributions
to the fine-structure.
One is-- kinetic energy is p square,
but now there is a higher order relativistic contribution,
p to the 4.
Then, there is a term, spin orbit coupling term.

And then there is the third term which we will
identify as the Darwin term.
Laplace operator e square over r.
So we have a kinetic energy correction.
We have a spin orbit correction.
And we have a Darwin term.
And what we want to do in the next 45 minutes or so is
we want to look at each of those.

Any questions?