Okay, so let's start. Let me first remind you of what we did last time.

We just reviewed the basics of the harmonic oscillator.

And we learned once again about the annihilation and creation operators which take you between the different energy levels of the harmonic oscillator.

So these are related to the position and the momentum and in particular let me remind you of the formula connecting the position and the annihilation and creation operators which is that the position is simply equal to 8 plus 8 giga.

So this is a Hermitian operator and you multiply with the length scale and this length scale we call it x subscript zpf or zero point fluctuations and it just denotes the width of the ground state.

And the width of the ground state once more is equal to h bar over 2m omega tapping the square root.

Now in particular at the end we discussed coupled harmonic oscillators and we coupled two of the harmonic oscillators.

And we convinced ourselves that the Hamiltonian would in general have the following form.

The Hamiltonian would be the parts that are related to the individual oscillators A and P.

And so for the mechanical example you would get a coupling that links the two coordinates xA times xB.

And so if you write it in terms of A and B it looks like this.

And the crucial simplification that can be made often and which we convinced ourselves is valid if the coupling is relatively weak and the oscillators are close to resonance is the following.

That this contains four different terms, two of which are unimportant.

These are the terms that would say create both an excitation at A and B or annihilate an excitation both in A and B because they don't conserve the overall energy and so they are not resonant.

And so in the end you often can approximate this interaction term by something which only leaves the processes which look simple namely annihilating an excitation in B and creating an excitation in A which just means you transfer the excitation.

And this really corresponds to the physical process that you would also expect classically if you excite one of the oscillator motions then after some time because of the coupling the energy is transferred to the other one.

Okay, now in the following I just want to carry this through for many coupled harmonic oscillators so let's just imagine having a chain of these coupled oscillators.

So what we think of would be for example a one dimensional array of oscillators each of them say represented by such a circle.

And if you like you can think of them as optical cavities.

They will be coupled.

So then if we just count those with an index j and I want to introduce the harmonic oscillator operators for each of those oscillators and I will call them A j.

So we can write down the Hamiltonian and let us just assume that each of these oscillators by itself would have a frequency omega j.

So I can easily write down the part of the Hamiltonian that does not yet include the coupling and it would be just a sum over all j h bar omega j times the number of excitations in this particular oscillator.

So that would be A j j j j.

Now you want to write down the coupling in the form that we already saw up there.

So you just transfer excitations along the chain.

And a typical term would look like this. You take out the excitation in oscillator j and you put it into oscillator j plus one.

And then of course there is also the reverse process where the excitation travels the other way.

And then you have to prescribe an amplitude for this to happen. You could even make this dependent on j but for the moment let us just assume that this amplitude is fixed.

Let us call it g and g would then denote the frequency at which excitations are transferred.

So you just sum this over all j.

Now you could go ahead and do the same for a two dimensional array or for a more complicated arrangement.

Also for arrangements where you don't only transfer the excitation to the nearest neighbor but rather transfer the excitation to a harmonic oscillator which is far away because there can be forces that are rather long range.

If you want to write down the most general version of the Hamiltonian for this collection of harmonic oscillators, and this is simple enough, it should be a bilinear form in these aj and aj dagger.

And we want it to conserve particle number so it should always have the combination a and a dagger. Never say two operators a dagger.

If the general version is easily written down, h would be in general a sum over two indices, namely any pair of oscillators can be coupled.

So let me sum over l and j. And then let me destroy an excitation in oscillator j and create an excitation in oscillator l.

And let me call the amplitude for this to happen h tilde. And you will see why I call it h tilde in any way. It has the dimensions of an energy.

But it is an object completely different from the Hamiltonian because this Hamiltonian lives in a many particle Hibbert space which has all the different combinations of excitations in the various oscillators.

Whereas this h tilde is just a matrix and it has the dimensions given by the number of oscillators. So it is a much simpler object.

And we would identify it as a single particle Hamiltonian for the bosons in this oscillator chain.

Okay, so now you can ask several things. One of the questions would be what is the time evolution of the a operators under this Hamiltonian?

That is a reasonable question. The other question would be how do I diagonalize this Hamiltonian? How do I bring it into a simple form?

So let's first write down the time evolution. This is simple enough. So we just write down the Heisenberg equation of motion.

Say for the operator a n, we know that we can obtain this by taking the commutator with the Hamiltonian.

Now this requires a little bit of work but since you know the commutation relations of the a operators, in particular a, aj equals 1,

you convert this out and you find that the result is again only linear in a operators and it's just given by a sum over all j where l is fixed as an index h tilde lj aj.

So it has a very simple structure and in principle you should know this structure if you followed the quantum mechanics one lectures

and you ever saw the Schrödinger equation for evolution on a lattice because this is nothing but some form of Schrödinger equation.

However you have replaced what had been the complex amplitude psi j, that is the entries of the wave function expanded in such a basis of orbitals so to speak,