So good morning everyone and again thank you for the home task and those who didn't give it to me but want to give it to me give it do it before the end of the lecture because at the end of the lecture I will give my comments about this home task.

So last time we stopped at introducing the position and momentum variables and then we introduced some variables that are combinations of position and momentum and the home task was to find their dimensionality.

I will write it once again. It was minus i. If I'm not mistaken. Let me write it down accurately.

The way they were defined it's I find it quite interesting that the way they were defined L cubed the size of the quantization volume H omega k and then there was omega square root of epsilon naught electric field plus one over square root of mu naught magnetic field k.

And the task was to find the dimensionality of this. So someone understood it a little bit incorrectly because there was also an expression through position and momentum and through position and momentum it's easier because here you have electromagnetic units and there you have just usual energy units divided by energy units here.

Anyway so these amplitudes and we did it in such a way that the Hamiltonian was given by the sum over modes of single mode Hamiltonians and single mode Hamiltonian had the same form as the Hamiltonian over as an energy of a harmonic oscillator looks.

So it's one half of Q k squared omega k squared plus p k squared. Yeah and then there at this point we stopped.

But today we will introduce the operators and that will be the most important step. It's already lecture six. It's halfway almost halfway through the course and then we now we announce that actually these Q and P are not just position and momentum but position and momentum operators.

Namely we pass from P k to P k with a hat and Q k to Q k with a hat which so far means nothing.

But then we announce that they obey so-called commutation relations. So in quantum mechanics it doesn't it is not it is not the same to write Q times P or P times Q.

So this combination the difference of Q times P P times Q is called the commutator of Q and P.

The commutator and it is postulated that this commutator is equal to I h bar the Planck constant.

And after this we rewrite the Hamiltonian in terms of these operators and it's going to change a lot of things.

So the Hamiltonian now will be the energy and this energy is again the sum overall k 1 over 2 omega k Q k squared plus P k squared.

So now I write hats and most important is also this operator that appears when I when I introduce when I when I transform the amplitude a k into an operator.

So I have to I have to write here is definition in terms of position and momentum.

That's what we did at the last lecture. So it's 1 over 2 h bar omega k.

If I copy it to be sure. Yeah. And here omega k Q k plus P k.

That was I P k plus I P k. So we say that Q and P are real quantities and then this amplitude is a complex variable.

So now I want to introduce an operator A and it will be just 1 over 2 h bar omega k omega k Q k now operator plus I P k operator.

And in quantum mechanics yes all observables correspond to operators.

So instead of observables like position momentum electric field magnetic field we now have operators.

But if the observables are real then the corresponding operators are Hermitian Q and P position and momentum operators should be Hermitian operators.

And what is Hermitian? Hermitian means that Q dagger. Yeah. Dagger means Hermitian conjugation.

If you again you probably some of you as I heard didn't study quantum mechanics but you can think of an operator as a matrix.

And a Hermitian conjugate of a matrix is a matrix that has complex conjugation and transpose.

So like you know what is a matrix. Yeah. So it's a complex conjugated and also transposed.

So the elements this element becomes this element and so on. That's what transpose means.

So probably you can understand what is the Hermitian conjugated conjugate of an operator and Q dagger is just equal to Q.

That means that the operator Q is Hermitian. However and the same for P.

However this A operator and I'm a little bit ahead of the lecture but I will tell you from the very beginning that it will be called photon annihilation operator.

And we will soon very soon see why it is called photon annihilation operator.

But it's clear from this notation that it's not Hermitian because if we do the Hermitian conjugation then Q dagger will be the same as Q.

P dagger will be the same as P but there is this I and so A dagger is not equal to A.

So it's non-Hermitian.

And now what we want to do is to write Hamiltonian in terms of A and A dagger operators.

And to do it we just write a combination what it means A dagger A.

By the way the hats over the photon annihilation operator and this one is called photon creation operator are usually not written.

So I will omit the hats from now on.

Let's write A dagger A in terms and of course we do it for the mode k.

So we just substitute this expression we'll get 1 over 2 h omega k and then we write the expression for A dagger.

It means that we have to put minus I omega k Q k minus I p k and these are operators right.

And then the Hermitian conjugate so A itself omega k Q k plus I p k.

And then we just open the brackets taking into account that p k times Q k is not the same as Q k times p k.

So it's 1 over 2 h bar omega k and here there will be omega k squared Q k squared.

Then there will be this term plus p k squared.

Already we got something like the Hamiltonian.

But then there will be the cross terms omega k Q k p k with the I so plus I omega k Q k p k.

And then the other term and it will have minus I p k Q k.

So we can we can drag the I out of the brackets and then minus p k Q k.