M1L2d.txt

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# Captions for 8.422x module
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Enough of classical physics.
All we have done so far-- and sorry for repeating it,
but I think I can't repeat it too often with so many
complicated equations on the screen-- all
we have done so far is we have rewritten Maxwell's equations.
We've rewritten Maxwell's equations
in Fourier space with the vector potential,
eventually with normal modes.
That's all we have done.
But now what we are doing is we do
this step in quantum mechanics, which you have already
seen a few times.
And this is you have written equations in such a way
that they look harmonic oscillators.
And then you postulate that the classical quantities
become operators.
The transverse vector potential and
the conjugate canonical momentum fulfill now commutators.
And we use those commutators to define them as operators.

Now, it looks particularly easy when we use normal mode
operators because the normal mode operators
after quantization become our a's and a daggers.
So I just want to sort of-- that was one of the things I wanted
to show you, that you can go through everything introducing
classical normal modes, and then the normal modes
turns into operators.
And now you have all your creation
and annihilation operators.
So in other words, quantization cannot be rigorously proven.
I mean, you cannot prove quantum theory from first principles.
You can have a mathematical framework
and then check it against nature.
And what we have done here is to formulate the quantum theory.
At this point, we've made a postulate.
To We've made a postulate that we have operators which
fulfill a commutation relation.
And it is now your choice if you want
to formulate, postulate the commutator
for the transverse vector potential
and its conjugate momentum.

Or if you immediately want to jump at the normal modes,
and that would mean you have the commutator for a and a dagger.
OK.
So we started with the electromagnetic fields.
We went through transverse fields,
vector potential, normal modes.
Now we quantize.
But now all the equations, all the substitutions we have made,
we can go backward.
So therefore we can now express the converse vector potential,
the electric and the magnetic field, by the normal modes.
Or that would mean in quantum mechanics after quantization
we can express them by the a's and a daggers.
And this is how we define the operator
of the electric and magnetic field in the vector potential.

So now our fully quantized theory
has operators, a or a dagger, or if you want,
now the first defined operators of vector
potential electric and magnetic field.
And just to remind you, since atomic physics is
between fields and particles, we have exactly-- we
have the standard definition of the particle operators.
Each particle is described by its momentum and its position.