M2L7m.txt

#
# File:   content-mit-8-421-2x-subtitles/M2L7m.txt
#
# Captions for 8.421x module
#
# This file has 224 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------

Let's get started with the Lamb shift.
Just to make sure that you're not getting confused,
there is also a lamp shift.
This is actually an old-fashioned word
for an AC Stark Effect.
People in the '50s and '60s found
that spectral lines in a mercury discharge
shifted when you took another mercury lamp
and illuminated the mercury with a strong light of another lamp.
So a lamp shift is simply a shift
due to the electromagnetic radiation emitted by a lamp.
But that's not what we're talking about.
We're going to talk about the Lamb shift which
is named after Willis Lamb who received the Nobel
Prize for the discovery that the 2S one-half,
and the 2P one-half level in hydrogen are not degenerate.
And he wouldn't have gotten the Nobel Prize
if it would have been simply about one little feature
in the spectrum of hydrogen. But the discovery
of this lifting of the degeneracy,
was actually opened up the field for the development of quantum
electrodynamics.
So this was the experimental discovery
which led to quantum electrodynamics,
and this is why it's a very famous effect, done by a now,
very famous man, Willis Lamb.
So, the full description of the Lamb shift would require QED,
and later in this course, we are developing the tools
to do that, I don't want to do it here,
I want to give you a simplified physical picture, which
actually nicely relates the Lamb shift to the Darwin term which
we have just discussed.
But sort of just in full disclosure, what you have to do
is, if you fully quantize the electromagnetic field,
you have a vector potential, which describes the vacuum mode
and you have a vector potential, which
is the operator of the fully quantized field.
And if you now carry out-- if you now carry out
second-order perturbation theory,
in this operator A of the quantized electromagnetic
field.
In other words, you allow the atom,
or the electron in the atom, to couple to all the empty modes
of the vacuum.
Then you obtain the Lamb shift in its full beauty--
and this is discussed nicely in Atom-Photon Interactions by Claude Cohen-Tannoudji et al.

So this is the nature.
The nature of the Lamb shift is a coupling to the vacuum modes.
But I want to capture that now in a semiclassical picture.
So the picture I am presenting here,
is due to Welton and Viki Weisskopf.
And yes, it has to address the nature of the Lamb shift,
and this is the coupling of the electron to the vacuum,
but the vacuum is not empty, the vacuum
is filled with the zero-point energy
of the electromagnetic field.
What we need is, that we have electromagnetic modes,
and they have a zero-point energy.
And each mode has a zero-point energy of h bar omega.
And maybe before I do any equations,
I should just give you the physical picture.
I think it's much clearer to understand that.
so the moment each mode has an energy,
then each mode has a fluctuating electric field.
Now what we have to do is, we have
to multiply the contribution, the energy of each mode,
by the density of modes.
We have to sum over all modes.
But then, in the end, what we get
is an expression for the fluctuating electric field.
And remember, in the Darwin term,
we assumed that the electron is trembling by itself,
and is smeared out over the Compton wavelength.
But now in addition to its own trembling motion,
the electron is now shaken by the electric field
of the vacuum, and this leads to an additional smear out,
and this is the Lamb shift.
So, that's what we want to put together.
So we want to calculate an additional contribution
to a trembling motion of the electron,
but this time it is driven by the electric field, which
persists even in the vacuum.
So that's the Lamb shift.
OK, so it's one-half h bar omega, per mode.
We have to use the density of modes,
and the density is per unit volume, and frequency interval.
So we have a density, nu is the frequency,
and you probably remember that the density of mode
scales with frequency squared.

That means that zero-point, the density
of the zero-point energy is one-half h nu per mode,
times the mode density, and this means
it's scales now with nu cubed, the frequency cubed.
And since we want to introduce now
the electric field, the energy in the electric field,
if you go from electric field to energy,
the energy density of the electric field
is proportional to e squared.
So therefore, what we derive from this picture,
that the vacuum is filled with an oscillating electric field.
And this oscillating electric field
is characterized by an value, by a spectral density e squared,
which is proportional to nu cubed,
and if I collect the constants h and speed of light, it's this.
So the question is, what does this electric field
do to the electron?
Well, let's not discuss the electron in a hydrogenic orbit.
Let's rather discuss, in a simplified picture,
what is the effect of such a field on a free electron?
And we will later discuss, that for very high frequencies,
an electron can be regarded as free.
So therefore, if s is the coordinate of the electron,
and you drive it with an oscillating electric field,
at frequency nu, then the driven amplitude of the electron
is the electric field.
I just want to make sure we keep track of the powers of nu.

Because we take the second derivative, acceleration,
we get two powers of nu, so we have an extra nu squared term.

Of course, the phase is random, so we're not
interested in the amplitude, we are
interested in sort of an average amplitude squared,
or an RMS amplitude.
Which is, the amplitude was proportional
to the electric field.
The amplitude squared is proportional to the square
of the electric field, and our prefactor involves, now, nu
to the four.
Just in the equation at the top, we
had an expression for the spectral density
of the electric field e squared, which goes nu cubed.
So that means now that the spectral density
of the motion of the electron s nu squared, goes as one over nu.
And yeah, e squared, h bar, pi square, m square, e cubed.

And now, yes, I've wanted to do it
in the same lecture as the Darwin term,
because we know that there was a change to the Coulomb potential
in the Darwin term, which came because the electron was
smeared out over an area s squared-- remember this was
the Compton radius squared-- and it involved the Laplacian
of the potential-- Taylor expansion,
second order, second derivative in three dimension,
is the Laplacian-- and for the Coulomb potential,
this is non-vanishing of course, only at the origin.

So therefore, we have now executed
what I discussed before.
We calculated the sort of displacement
of the free electron driven by the vacuum
fluctuations of the electric field, and these smear
out of the electron leads to a change
of the the average Coulomb potential, the average Coulomb
potential over the fast motion of the electron.
And that means now that we get a change of the binding
energy of the electron, which is nothing else-- in perturbation
theory-- then the matrix element of the perturbation operator,
but this perturbation operator was a delta function, we still,
and we have the prefactor, which is
the spectral density of the displacement s nu squared.
So let's wrap up.
So we know, because of the delta function,
it only affects s electrons.

Remember, I discussed at great lengths, that for an electron,
with principle quantum number n, the probability
to be at the center was n cubed.
So here, again, we have the n cubed factor
which we discussed for the hydrogenic wavefunction.
And the only thing which is no nontrivial, we have to do
is, we have to integrate over all modes nu.

So therefore, what we had so far was
the shift caused by one frequency of the electromagnetic
spectrum, but now we have to integrate over all frequencies.
So what we have to do is, our spectrum
s square goes as one over nu.
And if you integrate one over nu over all frequencies,
you get a logarithm, you get logarithmic divergences.
So therefore, our result is-- and I'm
using now atomic units alpha cubed,
Z to the four, the n-cubed factor,
everything is in units of atomic units,
the atomic unit of energy is one-half hartree
or two Rydberg.
But the factor which is now non-trivial
is from the logarithm, we integrate,
and we have divergences at both ends.
So we need a cutoff at the minimum
and at a maximum frequency.
For the maximum frequency, well, there is an natural cutoff.
We have to cutoff at the rest energy of the electron,
otherwise we are no longer doing single particle theory.
And the cutoff, the lower cutoff, is the following.
We've used a picture here for a free electron,
but we can regard the orbiting electron only as
a free electron on timescales, which are much faster than one
orbital period.
So therefore, we have to use, as a lower cutoff, the frequency
of the orbiting electron.
Which is Z squared, n cubed in atomic units.
So with that-- a little bit handwavingly, but it appears
in a logarithm anyway-- we have gotten
for the ratio of the upper and the lower cutoff
an expression which is n cubed over z squared times alpha
squared.
And if you apply it to the 2S state,
where it most important because there
is the degeneracy between P one-half
and P three-half to be lifted, we get a result
that the energy splitting is now, and with the assumptions
we have made, the logarithm is 8 over a squared, alpha squared,
and from that, we would have derived
a Lamb shift of 1600 megahertz.
Well, the exact value is 1,058 megahertz.
I'm missing five minutes to discuss the result in terms
of vacuum polarization, but that's
something we do on Monday.
Any quick question?
OK, great. See you on Monday.