Okay, let us start.

According to the lecture notes, we are now finishing lecture 6 and I will start lecture

7, but I will not finish it.

Now I understand that the total number of the lectures will be 13 and not 12 definitely,

so we will finish something like in at the beginning of February.

Yeah, this year will be today's lecture, then another two lectures and then the rest will

be in the new year.

All right, so continuing what we started last time, we were discussing different states,

quantum states and related operators because some of the states are defined as eigenstates

of a certain operator, all of them actually.

And we will continue further.

So we stopped it, let me see, I think we discussed coherent states and fock states and then we

not derived but wrote a commutation relations for the quadratures and then I wrote the uncertainty

relations that follow from these commutation relations, I didn't derive them, but I didn't

probably have time to discuss the meaning of uncertainty relations and this is important.

And the meaning of uncertainty relations is there are different physical interpretations

of this.

So first of all, for instance, x1 and x2, the quadratures do not commute, yeah, so it's

the commutator was, oops, I always forget these numbers, so q and p is iH and here it's

i over 2, yeah, i over 2.

So if they do not commute it means that there is uncertainty, so the delta x1 delta x2 is

larger or equal to, remember there was a formula, I should take one half times the mean value,

the absolute value of this, so it's one-fourth.

And it means that I cannot measure them simultaneously.

It means that if I try to measure them simultaneously I will always have some uncertainties, right,

but I can try to measure them simultaneously so that the uncertainty in one value is very

small but then the uncertainty in the other value is very large, becomes very large, the

product being limited.

And this gives you already a hint of the phenomenon of squeezing which we will discuss later.

So it's possible to measure one of them with very high accuracy, so like delta x1 is small

and delta x2 then will be large.

And then, but also there is an important interpretation of this if I try to introduce the probability

distribution, joint probability distribution that one quadrature has a value x1 and the

other quadrature has a value x2.

In principle it's a normal thing in mathematics, in physics, joint probability distribution.

We'll see that it is also impossible because I mean I can introduce in many different ways,

I can introduce some probability distribution but I always have to face the price for assuming

that there is such a point x1, x2 and the price for this will be that the probability

distribution will turn out to be negative for instance for some cases or singular for

some cases and that's why all these probability distributions are called quasi probability

distributions because they are not really probability distributions and this we will

see in the next, in the coming next lectures.

So another thing that I wanted to add concerning the simultaneous measurement, imagine that

we measure one of them very precisely and then measure the other one and the idea is

that the measurement of one of them very precisely disturbs the other one and makes it spread

like uncertainty makes it increase.

So these are three different interpretations of the Heisenberg uncertainty relation.

And now I want to slightly discuss a different, to discuss a slightly different subject and

namely as long as we have this photon number operator I want to, which is I will remind

you a dagger a.