Okay, so it's time to turn our attention again to the field because we started with a coupling

between the field and the qubit and we learned for example that you could use this for a

qubit readout and then we focused on the qubit and learned about quantum computation.

Now I want to focus again on the field mode and so what we want to discuss today is how

to describe the quantum states of a harmonic oscillator and what are the most interesting

quantum states that we want, the simple coherent states that we already learned about.

And so really when I say quantum states of the field we will be dealing with quantum

states of a single harmonic oscillator that is a single field mode.

Now the first thing we want to discuss is a description that is very well suited and

that treats the position and the momentum on equal footing and that is called the Wiegler

density.

The idea behind the Wiegler density is that it would presumably be nice if you were able

not only to give the probability density in terms of position and the probability density

in terms of momentum but like in classical mechanics would be able to write down a sort

of probability density both for position and momentum.

So why is that desirable?

Well for example if you want to discuss the classical limit at high temperatures hopefully

this kind of density which we have to define would turn into the classical phase space

probability density that it would arrive at in classical statistical physics and so you

could easily learn how quantum mechanics turns into classical mechanics at high temperatures.

Now at first sight it seems quite hopeless because of course quantum mechanics is built

on the assumption that you cannot measure simultaneously position and momentum so either

you can calculate the probability density for position that's just psi of x squared

or else you can calculate the probability density for the momentum that you obtain by

just transforming into the momentum space and then again taking psi squared in momentum

space.

And so it's not clear how we could at all arrive at a joint probability density for

x and p and as we will see the density we are going to arrive at is not quite a probability

density because it can become negative at some points but that is the only price we

have to pay.

So one of the easiest ways to see how one should define such a density is the following.

Imagine you have a wave function given a position space but now you just want to transform into

momentum space to evaluate the probability density in momentum space and so the way you

would do that is just take the Fourier transform of your wave function position space.

So there needs to be a minus i here because you know the plane weights would have e to

the plus of kx and then you have to worry a little bit about normalization and the proper

normalization actually turns out to be square root of 2 pi h bar and the denominator just

to make psi tilde of p squared integrate to 1.

So that is the wave function in momentum space and then you can immediately calculate the

probability density in momentum space.

You just take the square of this.

Now taking the square of this I can as well, I will get two integrals or two integration

variables and one of them I will call x1 and the other one I will call x2.

So that would be the square of this and that would be integral dx1 dx2 over 2 pi h bar

e to the i p over h bar x2 minus x1 and now you already see that x1 is the variable where

I didn't take the complex conjugate so I should go with psi of x1.

The x2 since it has a different sign in the exponent obviously is the variable where I

chose to take the complex conjugate so it should go with psi star of x2.

Now you see that this only depends on the difference in coordinates so I might as well

transform to other variables where I take the average position and the difference.