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Good morning, welcome to lecture number 16, which is titled the Quantum Harmonic Oscillator,

and usually this is something like lecture two.

But in fact we will need most of what we developed so far in order to really understand the Quantum Harmonic Oscillator.

Now, language can be misleading, particularly in a technical subject, and if we talk about the Quantum Harmonic Oscillator,

putting this adjective quantum in front, somehow suggests there's something like a harmonic oscillator,

and now we look at a particular type of harmonic oscillator, which has an additional property being quantum.

Okay, but in fact as I remarked before, in the world there are only quantum elephants,

and thus quantum is not a property a thing can have or not have, it's just something that is always the case, okay.

So there is no such thing as a classical harmonic oscillator in the real world.

Of course in the textbooks there is, right, and in the exams there is, but in the real world there is no such thing as a classical harmonic oscillator.

So, and only quantum harmonic oscillators. Well, this actually relieves us of a duty, namely of the duty to explain how you start with a classical oscillator,

and how you arrive at the quantum oscillator. So we will thus not entertain any type of quantization ideas

that start from a classical system and aims at constructing the quantum system from that.

We will thus not entertain any such type of quantization scheme, because it's meaningless in the first place.

It's not like, oh, you really have the classical system, now you make a quantum system out of it. This is not what happens.

So it is rather the classical physicist, so me two semesters ago, it is rather the classical physicist who has to explain

what the omission of the adjective quantum is supposed to mean.

So with full justification I could call quantum harmonic oscillators, I could just call them harmonic oscillators,

and of course they're quantum, and the classical physicist has to say, well, I want to, under these and those circumstances,

I want to make this and that approximation, and so I rather speak of a classical, or a classicalization of the quantum system.

Okay, so this is the philosophy of what we're doing for very good reason.

And although I just said we will not entertain any such type of quantization, I will now briefly do it anyway.

But this is the program, and this is just a brief deviation from it.

So what people say. So let's briefly consider why it is so problematic to think of what I say we wouldn't think about.

So people say, I mean people is always the others, right, the man on the street that's never yourself.

So people say, under certain circumstances, this is also in oral exams, you often get the answer,

under certain circumstances which is supposed to cover up for all the sins of omission that follow, well it doesn't always work,

so people say under certain circumstances one gets or obtains the quantum system that corresponds to a given classical system.

By the following rule.

So what is said is that if you have a classical observable, you remember what that is, a classical observable is an object F that depends on Qs and P's,

that depends on the position and the momentum, the position and the momentum of the system at a certain point in time.

I mean this is already in coordinate version, more strictly speaking it's a function on the phase space,

which in particular circumstances may be the cotangent bundle of the configuration space, blah blah blah.

Anyway, a function of the Qs and the P's and the continuous function, if not several times differentiable function,

but you also know that the quantum observables are quite a different beast, and ignoring all domain issues or choice of Hilbert space, whatever,

there is the idea that you can translate this to a quantum observable, well I should write the classical observable with a small f,

and then the quantum observable can be written like this, where you have to replace obviously, well it's rather you use the same function,

where you replace the Q, which itself is a function of the Qs and the P's, you replace it by the quantum mechanical position operator,

which we defined before, and the P's you replace by the quantum mechanical momentum operator,

so if you have several then you have the Qi's, if you have several you have the P sub i's, you do this, you replace the Qs by those and the P's by the others.

Okay, now of course there are few people who say this is without problems, and I will point out the problem,

but as I will show you the problem is so severe that it seems almost ridiculous to start with it in the first place.

So the idea is if I show you for instance the energy observable of the classical harmonic oscillator, do you remember that one?

What is the Hamilton function, the Hamiltonian function of the classical harmonic oscillators of mass m and angular frequency omega?

So e.g. classical harmonic oscillator energy observable, the thing is called little h, depends on the Qs and the P's,

and it's given as 1 over 2 mP squared minus m omega squared Q squared, so this is classical,

so it's the classical Hamiltonian for a harmonic oscillator, mass m angular frequency omega.

Now we apply our rule, and I just lay it down, and you come to the quantum harmonic oscillator energy.

And well, so apparently I now have a Hamiltonian which I can call capital H, but it's actually to be constructed as the H of the Qs and the P's,