M1L3e.txt

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# Captions for 8.422x module
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If you just want to think about it intuitively,
alpha is like the best description
of an electromagnetic field.
So if you write down this, we ask,
what is the probability for an arbitrary quantum state
described by an arbitrary statistical operator?
What is the probability that the electric field is alpha
in the complex description?
So that's what those quasi probabilities are, q of alpha.
And we immediately looked at some examples, of course,
of probabilities.
We showed that the vacuum state is sort of an area,
it's a Gaussian, and the area is-- I don't know,
I forgot the prefect are.
It's on the order of one or one half.
We realize that thermal states are also a Gaussian, singled
at the origin.
It is a much wider distribution so thermal states
have much more uncertainty in the electric field
than the vacuum state.
And then we looked at the coherent state
and now, of course, when we looked at the coherent state,
we realize that coherent states have a maybe not as wonderful
as I try to make you believe.
They have some nasty properties and that
is when you ask, what is the probability,
the cause of probability of the coherent state to be beta?
Now if you would expect a delta function because when you ask,
what is the probability of a momentum state to be p naught?
If you have a momentum state p naught,
it has a delta function at p naught.
But here it's not a delta function,
it's a Gaussian of standard deviation.
I think the standard deviation here is-- what is it?
1 over square root 2?
Because the Gaussian has usually 2 times sigma squared
and since the denominator is unity,
so the standard deviation of 1 over square root 2.
And that showed us that the coherent states are not
orthogonal.
You have to be a little bit careful,
it's not your standard basis set.
It's an over complete basis.

So therefore, the coherent state is not
a delta function in the course of probability,
it's a little bit blurred circle with an area
on the order of unity.

You have some homework assignment to look at it.
So in some way, how you should look at
it is that if you see some quasi probability distribution,
the distance from the origin-- this
is the absolute value of the electric field.
The phase here is almost 45 degrees,
but if you look at this uncertainty,
you would also say, well, the phase is uncertain to
within this angle.
So you can pretty much read the uncertainty
in the total electric field, the uncertainty in the phase,
you can read pretty much everything
you want from this diagram.

Now when we look at a number state,
[INAUDIBLE] off know in quantum mechanics,
number and phase are complementarity.
If the number of photons is fixed,
you know nothing about the phase.
And indeed, the cause of probability of a number state
is rim.
It has no phase.
It has completely random phase over the 2 pi circle.
The energy is sharp of a number state
since the energy is e squared.
You may have expected a delta function in the radial called
in it, but what you get is also something
blurred on the order of unity.
And I want to say something about that in a second.
Finally, we discussed the time dependence and the time
dependence is very easy.
After all, we're dealing with an harmonic oscillator.
And in an harmonic oscillator, if you
have a plane of x and p, symmetric and anti-symmetric
combination of a and a [? daga ?] in this plane,
in a harmonic oscillator, the quantum state
is just rotating-- circular notation with omega.
And, indeed, we showed that when we apply the time evolution
operator-- and some of you replied,
of course, with a minus sign, it moves
with e to the minus i omega t and, therefore, everything
rotates in a clockwise way.

Now we discussed the operator of the electric field
and in this quasi probability-- sorry, we
discussed the operator of the electric field
and I hope you'll remember that in the analogy
with the harmonic oscillator, the electric field was
a minus a [? daga ?] and this is the momentum operator.
In those quasi probabilities-- and we will see more about it--
something which is sharp in momentum
is a sliver parallel to the x-axis.
So therefore, you can regard the vertical axis,
which is an imaginary part of alpha,
as the momentum axis and you can regard
this a the horizontal axis as the x-axis.
So therefore, since momentum is electric field,
you always get the electric field
by projecting onto the vertical axis.
And if you put check this fuzzy ball,
you get a value 0 with some uncertainty.
And if this quasi probability starts
to rotate due to the time evolution,
we get an oscillating electromagnetic field,
almost classical, except for the fuzziness.

So that's where we want to continue.
Any questions about that?
Yes?
Why's there no phase fuzziness in the [INAUDIBLE]?

No phase fuzziness-- there is a phase fuzziness.
For instance, if you would say the phase is determined
by this 0 crossing, you don't know
exactly when the 0 crossing happened
and that corresponds to an uncertainty in the phase.
Trust me, everything is in this diagram.
Now there are two things we want to continue.
One is I want to show you that the coherent state is
a minimum uncertainty state.
The product of delta x delta p, delta p is just--
is it h bar or h bar over 2?
One of the two?
So it's a minimum uncertainty state
and, therefore, you can never have
a quantum state which is less fussy than the coherent state.
So this fuzziness here is the intrinsic uncertainty
of quantum physics so that's what you want to discuss today.
But then we will immediately start with non-classical states
and that is, well, if this area is determined
by Heisenberg's uncertainty relation, what can we
maybe deform the circle into an ellipse,
and these are three states of light.
So that's an outlook.
That's what you're going to do in the second half