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Welcome to lecture number 17. Today we'll consider measurements in more detail.

And you may recall that axioms 3 and 5 of quantum mechanics that are presented in the first lecture, axioms 3 and 5, they deal with measurements.

And so far we invested a lot of work into understanding axioms 1 and 2, well, into understanding the basic notions in there.

And then we had a few applications where we calculated the spectrum of angular momentum operators or of the Hamilton operator,

the energy operator of a specific system, namely the harmonic oscillator last time.

But before we continue with more systems, we of course want to know, well, once we did this calculation of the spectrum, what have we learned?

What does it have to do with how to interpret the system if we measure it in the lab?

And of course, so far we talked and we calculated the spectra and the spectrum of an observable,

and I'll emphasize this again during the lecture, tells you what are the possible outcomes of measurement.

What are the possible outcomes? But that's very little information.

I mean, in classical mechanics, the possible outcome is the whole real line.

Okay, it's not very much of information in quantum mechanics. It's like, oh, you could have these and those energies.

Well, which energy do you really have in a concrete system?

And it turns out, and that's what the axiom says and you probably heard before, you cannot say, you cannot predict.

Well, you can observe, you can do the experiment and the measurement apparatus will tell you,

10 joule is the energy, but you cannot predict.

You can only predict the probability that the outcome will be 10 joule.

And that probability will not only depend on the setup of the system,

it will also depend on the state the system is in.

So today we'll have to look again at what actually is a state of the system, what is a pure state, what is a mixed state,

what does it mean for measurement.

And what we still haven't done that, so far we dealt with axioms one and two, roughly speaking.

Today we deal with axioms three and five.

And what is still missing is what is usually discussed first.

And then we have axiom four, which is the dynamics of what people call the Schrodinger equation.

We put this to the very end.

We first want to understand if we have a system in a certain state, how do we understand that system?

And only once you understand all of that, then we see how the state evolves.

And then we have everything together.

Okay, so that is this.

In order to illustrate these axioms, because they're already written down and there's no more theory to be developed,

it's only to be illustrated.

So in order to illustrate these two axioms,

we will repeatedly refer to the example, or to the concrete system of the harmonic oscillator,

which we discussed last time, as an example, so that the explanations do not remain just in the abstract.

Okay, so let's look at section one on what we got for the harmonic oscillator.

And the section title is a spectral decomposition of H, so of the energy observable, for the harmonic oscillator.

As I said, this is of course only a special example,

but I want to recall the important stuff and to do the spectral decomposition of that in order to illustrate what follows in this lecture.

So recall that for the Hamilton operator of the harmonic oscillator, which we wrote down last time,

we found an orthonormal basis, an O-N basis of H eigenvectors, which we labeled psi n.

H psi n equals E n psi n, and this E n is just short for H bar omega times n plus one half.

Those were the possible eigenvalues where n was any non-negative integer, n in N0.

So this is what we found last time, and they were generated, so more precisely,

we started with a ground state, which we calculated from the theory, that was something like m omega by 2 pi H bar x

minus m omega by 2 H bar x squared, that was the ground state of the, so it's of x here, of the harmonic oscillator.

And then one obtained the higher energy states, psi n of x, by repeated application of the A plus operator.

So if we apply the A plus operator n times, so it's a composition, n times A plus,

you apply it to the psi 0 of x, and you provide for appropriate normalization factors, A n.