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M3L17a.txt
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M3L17a.txt
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#
# File: content-mit-8-421-3x-subtitles/M3L17a.txt
#
# Captions for 8.421x module
#
# This file has 131 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
I want to come back to the rotating wave
approximation revisited.
So I'll revisit the revisit of the rotating wave
approximation.
And sometimes when I have discussions
with students after class, I realize
that something which I sort of casually mentioned
is either confusing or interesting for you.
And there are two aspects I actually
want to come back here.
So several people reacted to that.
But some felt it was maybe a little bit too complicated,
or others asking about some detail.
So let me come back to two aspects.
And I hope you find them interesting.
One is when we sorted out all those terms,
those need to angle on momentum selection rules.
But I made sort of the innocent comment,
well, if you've already got minus omega in a time
dependent Hamiltonian, one term is responsible for absorption.
One is for emission.
And when more than one person asked me about it,
I think many more than one person in class
would like to know more about it.
So therefore, let me spend the first few minutes in explaining
why it is a time dependent term in the Hamiltonian
with plus or minus omega, why is one of them
responsible for absorption, and one
is responsible for emission?
Well, we have Schrodinger's equation,
which says that the change of amplitude,
the change of the amplitude in state one has a term.
And if you start out with population in state two--
that's a perturbation field we start in stage two--
then it is the only term where this differential information
for an off-diagonal matrix elements
puts amplitude form state two into state one.
So what I'm writing down here is just Schrodinger's equation.
And the operator V is the drive field connecting
state two to state one.
And so if I just integrate this equation for a short time,
between time t and t plus delta t, and I'm asking,
did you change the population of state one, which
is now our final state?
Well then, you integrate over that for time interval delta t.
But now comes the point that the initial state has,
in it's time dependent wave function, a factor which
is e to the minus i omega 2t.
The final state, which I called one,
has, because it's a complex conjugate, plus omega 1t.
And let's just assume we have here the proportionality to e
to the i omega t.
And let me just say omega can now be positive and negative.
It will be part of the answer whether it
should be positive or negative.
Well, this integral here becomes an integral of e to the i omega
1 minus omega 2 plus omega t, integrated with time.
Well, and this is an oscillating function where,
if you integrate it with over time,
it will ever reach to 0 unless omega
is equal or at least close to the frequency
difference between initial and excited state.
So actually, what you encounter here
is-- well, what I've derived for you here is actually,
you can say, energy conservation.
I didn't assume it.
It is built into the time evolution of the Schrodinger
equation that you can only go from omega 1, from state one
to state two or state two to one if the drive term has
a Fourier component omega which makes up for the difference.
Or-- I'm using different language
now-- if through the drive term, you provide photons,
you provide quanta of energy where
omega fulfills the equation for energy conservation.
And you also see from this result,
when omega 1 is higher than omega 2,
omega has to be negative.
When the reverse is true, omega has to be positive.
So that's why I said the e to the plus
i omega t term is responsible for absorption.
The e to the minus i omega t term
is responsible for stimulated emission.
You also see, of course-- but I stop here
because I think you have heard it often enough.
If you indicate over short time derivative t,
the equation has to be fulfilled only to within 1 over delta t.
This is sort of the energy time uncertainty.
For short times, you don't have to-- the photon
energy does not need to match exactly the energy difference.
And you also realize, when we think about omega
is close to resonance, then e to the i omega t does absorption.
But if you're in the ground state, e to the minus i
omega t leads now to a very rapid oscillation here,
which is close to it a 2 omega oscillation.
And we've discussed that in the context of the AC Stark shift,
that this gives rise to the Bloch-Siegert shift.
We've also discussed that this term
is the rapidly oscillating.
And it's nothing else than the counter-rotating term which
we usually neglect when we do the rotating wave
approximation.
So everything we have discussed in this context-- core
counter-rotating term, energy conservation,
Heisenberg's uncertainty, time energy uncertainty--
actually comes from this kind of formalism.
Any question?
Of course.
If you quantize the electromagnetic field,
then you don't have a drive term with i to the omega t.
You just have A and A daggers for the photons.
And the question, which term absorbs a photon
or creates a photon, does not exist.
Because you know it whether it's a or A dagger.
But you have the two choices, whether you
want to use a fully quantized field with photon operators,
or whether you want to use the time dependent formalism using
a semi-classical or classical field and the Schrodinger
equation.
This second comment I wanted to do
is using the semi-classical picture,
I was sort of going with you through some examples, when
the rotating wave approximation is necessary, when not.
When do you have counter-rotating terms?
And yes, everything I told you is, I think,
is the best possible way, how you can-- well,
I assume because I haven't found a better one-- the best
possible way to present it and explain it