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M2L11g.txt
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M2L11g.txt
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#
# File: content-mit-8-421-2x-subtitles/M2L11g.txt
#
# Captions for 8.421x module
#
# This file has 156 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
Let's now be specific.
Let's assume the electric field points
in the z direction for an isotropic medium.
Our dipole moment, the time-dependent dipole moment
which we induce, is also pointing in the z direction.
And so we want to calculate now, what
is the dipole moment which is created by the drive term,
by the driven electric field?
And for that, we simply use the perturbation field
we have just applied.
We take the ground state and its first order correction,
and calculate the expectation value of the dipole moment.
In the line at the top, we have the first order correction
to the ground state wave function,
and so we just plug it in and what we obtain
is result where we have the matrix element squared.
Remember, we do first order perturbation field,
which is one occurrence of the matrix element,
but now we take a second matrix element because we induced it
in the dipole operator.
So this gives us now a sum over matrix element squared.
We have e to the plus i omega t and e to the minus i omega t.
This means we get two times the real part of this expression,
and the time-dependent term is e to the i omega t,
and then we have the term with plus and minus omega.
And here, [INAUDIBLE] most importantly,
everything is driven by the electric field.
We can now write the result in an easier way.
We can use this result with omega minus omega.
We can write that as 2 times omega kg omega
kg squared minus omega squared times cosine omega
t times the electric field.
So we have the matrix element.
We have integrated the e to the i omega t function and such,
so what we have here now is the time-dependent electric field,
and what we have here is the factor by which we multiply
the electric field to obtain the dipole moment,
and this is the definition of the now
time-dependent or frequency-dependent
polarizability.
This part here can actually-- that's how we often report
and that's how you often find in textbook the result,
the frequency dependence of the AC polarizability,
but I like to rewrite it now in a different way which
is identical, and it shows now that there
are two contributions.
Those two contributions, one has in the denominator,
let's assume we excite the system close to resonance.
Omega is close to omega kg.
Then one term is much, much closer.
[INAUDIBLE] to a [? real ?] resonant excitation,
and from our discussion about rotating frames,
we have all this [? real ?] resonant excitation corresponds
to a co-rotating term, and the other one corresponds
to a counter-rotating term.
We've not assumed any rotating fields here,
but we find those terms with the same mathematical signature,
and I will discuss a little bit later what the physics is
between the co-rotating and counter-rotating term,
and it's the co-rotating rotating term which
is the term which survives this so-called rotating wave
approximation.
I just want to identify those two terms,
and let's hold the thought until we have the discussion.
What I first want to see is the limiting case.
We have not made any assumptions about frequency.
When we let omega go to zero, we obtain the DC result.
It is important to point out there
when we have the DC result, we can only
get the correct results because we
have equal contributions from co- and counter-rotating terms.
So that's sort of a question one could ask,
which mistake do you do for the DC polarizability
when you do the rotating wave approximation?
Well, you miss out on exactly 50% of it
because both terms become equally important.
I have deliberately focused here on the calculation
of the dipole moment because the dipole-- I simply calculated
the dipole moment as being proportional
to the electric field, and the coefficient in front of it
is alpha, the polarizability.
You may remember that when we calculated
the effect of a static electric field,
we looked for the DC Stark shift,
for the shift of energy levels.
We can now discuss also the AC Stark shift, which
is a shift of the energy levels due to the time-dependent
field, but I have to say, you have
to be a little bit careful.
And sometimes when I looked at equations like this,
there is the moment of confusion what the question
is because the wave function is now a time-dependent wave
function.
It's a driven system.
It's no longer your time-independent Schrodinger
equation and you ask what is the shift
in the value of eigenvalues.
So the AC Stark shift here is now
given by the frequency-dependent polarizability.
And then, and I know some textbooks do it
right away and at the end of the day it may confuse you,
it uses an average of E square.
So in other words, if you have an electric field which
is cosine omega t and you calculate what is the AC Stark
shift, you get another factor of 1/2
because cosine square omega t time averaged is 1/2.
So anyway, just think about that.
It's one of those factors of 1/2 which is confusing.
Will, you have a question?
So when we take omega goes to zero from our previous results,
are we still justified in neglecting the transient term?
Yes, but why?
What will happen is the transient term is really
a term which has time dependence,
and even if omega is zero, just a step function
of switching it on creates an oscillation
in the atom at a frequency which is omega excited
state minus omega ground state.
You may think about it like this-- I give you
more the intuitive answer.
Take an atom and put it in electric field.
If you readily switch on the electric field,
you create a dipole moment by admixing at zero frequency
a P-state into the S-state, and that displaces the electron
from the origin.
But if you suddenly switch on the electric field,
you actually create a response of the atoms which
has a beat note between the excitation
frequency of the P-state and the excitation
frequency of the S-state.
And what you regard is the DC response of the atom
is everything except for this transient term.
However, and this tells me something
about the different formulas in quantum mechanics,
when we talked about the time-independent perturbation
field, we never worried about the switch
on because we just did time-independent perturbation
field and we sort of assumed that the perturbation
term had already existed from the beginning of the universe.
So it's not that we excluded the term.
We formulated the theory in such a way
that the term just didn't appear.
But if you switch on a DC field, you
should actually if you want an accurate description
to time-dependent perturbation theory,
you get the transient term even for a DC field,
and then you discuss it the way I did.
OK, so if you want, these are the textbook result.