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M1L2a.txt
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M1L2a.txt
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#
# File: content-mit-8422-1x-captions/M1L2a.txt
#
# Captions for 8.422x module
#
# This file has 148 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
Today we start from first principles.
So I want to give you a rigorous derivation of what
this course is based-- namely, the Hamiltonian
to describe the interaction between atoms and light.
I mentioned to you on last week that atomic physics
can be defined about everything what
is interesting about the building blocks of nature atoms
interacting with light electromagnetic fields, Coulomb
fields.
And today we talk about the first principle Hamiltonian.
The reference for that, and I'm closely
following that derivation, is the book
Atom-Photon Interaction, the green book by Cohen-Tannoudji,
Grynberg and others.
And it's taken from the appendix.
And if you flip through the appendix,
you will find out that there are about 100 equations.
And today we discuss every single of them.
[LAUGHTER]
However, the good news is the result
in the end is simple and intuitive.
In the end, we have what we want,
that the electromagnetic field couples different energy
levels.
And then we can play around with quantum gaze,
with laser cooling.
All we need is a coupling matrix element.
And for most of the course, I will not even
elaborate what this matrix element is about.
But here we derive it from first principles.
The result we will use most often
is actually the electric dipole approximation,
which I'm sure you have already seen many, many times.
So in that sense, I am-- I have sometimes second thoughts.
Should I dedicate a whole lecture
to derive something already?
On the other hand, this is sort of the meat of atomic physics.
And I want to go as deep as possible
into it that whenever you wonder what form of other interactions
exist, you have a reference to look it up.
And so to some extent I want to also encourage
you to read more about the fundamental nature of how light
and atom interacts.
I will give you another reference a little bit
later on.
But as a motivation, I want to tell you
what I learned when I prepared this lecture.
And I hope that there is something for you
to learn from this treatment.
Well, one is a rigorous separation
of local fields from radiation fields.
This is-- let me just write it down-- rigorous separation
of local Coulomb fields from the radiation field,
so to [? scalely ?] separate the terms in the Hamiltonian.
Let me explain that.
When we introduce electromagnetic fields
into our world, we would say we introduce them
by saying particles have charge.
But some parts of the charged particle physics
is something we want to sort of include
in the description of atoms.
And that's the Coulomb field of photons.
The structure of atoms is electromagnetic.
But when we now introduce the electromagnetic field
as a new degree of freedom, photons
can be emitted, absorbed and such,
we only want to talk about the electric field which
belongs to the photons and not the electric field which
belongs to the atoms.
And you may wonder, since electric fields,
can you distinguish between two kinds of electric fields?
The answer is yes.
And I want to show you how mathematically you distinguish
when you introduce electromagnetic fields
between the fields which stay with the atom, which don't have
an extra degree of freedom, and if you
do canonical quantization, they don't enter
as an extra variable, and the fields
which are the photons and these are the extra objects
we have to consider.
What is also related is when you do second quantization.
If you wanted, I could quantize the electronic field for you
in five minutes.
Just say it's a harmonic oscillator,
you're familiar with harmonic oscillator,
and it's just a word.
But yes, and this is correct, and I will remind
you of that little bit later.
But there is one non-trivial aspect
if you want to do quantization of a field theory.
You have to make sure that before you quantize it,
you know which degrees of freedom are really independent.
So the question is, how many independent fields does
the electromagnetic field have?
The naive answer is three electric field components,
three magnetic field components, that makes six.
But of course they are not independent,
and we will discuss that.
So we'll spend some time in the identification
of truly independent field components
or degrees of freedom of the electromagnetic field.
So all that is an excursion into classical physics,
because it is just reformulating classical electrodynamics
to be ready for quantization.
So actually, all the work is done
to discuss and derive the appropriate classical
description.
That means to eliminate all redundant variables
and then at the end have variables
which yes, will look like an harmonic oscillator.
And once we have reduced it to degrees of freedom which
look like a harmonic oscillator, it is very straightforward
to do the field quantization following the recipe
of the harmonic oscillator.
There's one more highlight, if I want to say so.
And this is a truly rigorous derivation
of the electric dipole approximation.
In most quantum mechanics textbooks,
you do in dipole approximation, and you wind up
with the electric dipole Hamiltonian, which
I have already put up there.
But you do actually unnecessary approximations
in terms of neglecting an A squared term.
And they are often confusing discussions,
what happens with the A squared term?
I want to show you a more rigorous derivation
of the electric dipole approximation, which
is including the quadratic term that actually means,
that's the take home message, the electric dipole
approximation is actually better than many of you
have thought so far.
Since I don't want to write down 100 equations,
I copied them and summarized them out of the book.
You'll find them all, as I mentioned before,
in the appendix of Atom-Photon Interaction.
But I sort of want to walk you through.
So what I'm aiming at is that you
have sort of a complete overview,
how is everything coming together?
And if it's just mathematical, well, mathematics
is always exact.
I can go faster.
But I really want to highlight here