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So good morning and welcome to lecture 19, the free Schrödinger operator.

Today we will be concerned with the Schrödinger operator for a vanishing potential.

And that is what we mean by the free Schrödinger operator.

So the Schrödinger operator for a vanishing potential,

or if you shift the energy of course for a constant potential,

this guy we call H3, because that corresponds to a particle that doesn't feel a potential, so it's free.

And that operator is minus h-bar squared by 2m times the Laplacian, the Laplace operator.

This is the operator for a free quantum particle of mass m, which is the energy observable.

I said it's the operator, what does that mean? It's the Schrödinger operator for that, which is the energy observable

for what we call a free particle for a free particle of mass m.

And so in this section we'll take the dimension of space to be 3, which is the standard in non-relativistic quantum mechanics of one particle.

We could also take it as 3n if we wanted to describe a constant number of particles in the problem.

I emphasize the constant because one also has opportunity to discuss particle creation annihilation, but that's not the idea here.

So then this directly translates, then the Laplacian is just one over the second, it's the sum of second derivatives in all these directions.

Anyway, so today we'll take n equals 1, but this is mathematically exactly the same.

So this is the physically simplest quantum mechanical system you can imagine, where you could say spin one half is even simpler,

but that comes with less intuition. So if you come from a classical point of view, you say I understand a classical particle of mass m

on which no force acts, that's Newton's first law talks about that, right, it's the free particle, it's in that sense an extremely simple physical system.

Now why does it only come up in lecture 19? Well, because in quantum mechanics, mathematically, this is by far not the simplest classical system.

We discussed this very favorite example of the quantum harmonic oscillator in a previous lecture,

and there we had a discrete spectrum, a point spectrum, all the elements of the spectrum were eigenvalues and so on.

Well, this is kind of easy and so on, so the harmonic oscillator, which in classical mechanics is more complicated than the free particle,

in quantum mechanics it's simpler because it doesn't use the full technology that we developed.

But as we will see today, in order to discuss this guy, its spectrum, which we will derive today,

and its time evolution, which we'll have a closer look at today, we need the full machinery we developed, right.

So the simplest physical system requires that full machinery, which again should reinforce the fact why we did what we did.

It was not in vain, it was not for generalities sake or something like this, or for the sake of being unnecessarily abstract,

it is exactly what you need to even discuss the simplest system.

Okay, so we study, or we will derive in today's lecture, the spectrum of H3,

we want to know what are the possible energies and study the time evolution of pure states.

Well, we could also study the time evolution of mixed states, but if we understand the pure states, the others follow.

Okay, now I said this is the energy observable, you know to be an observable it needs to be a self-adjoint operator.

Before we can talk about this being a self-adjoint operator, so if we want to make true on this statement,

we need to specify a suitable domain on which this is indeed self-adjoint.

So the first section, the first thing we have to clarify is what is the domain of self-adjointness of the such defined H3,

because otherwise we can claim this to be an observable, but in fact we do not really know.

So throughout today's lecture in order to lighten the notation a little.

In order to lighten the notation a little, we will choose units throughout

such that m equals one-half h-bar squared, which immediately implies that H3 is just minus,

this minus being very important, we don't want to scale that away, minus the Laplacian.

Okay, so you can always reinstate the, or you can just keep them, I don't want to carry them around.

So in the previous lecture on the Fourier operator, we saw that if you take the Fourier transform of a multiple derivative,

there was this multi-index, and we can make good use of that because we're in higher dimensions,

of a function that can be Fourier transformed.

So what we discussed that last time, that actually is minus i, no, it's ip multi-index alpha,

where actually by the i multi-index alpha, so you should write p multi-index alpha,

and the i is of course just the sum of all the components of this multi-index,

times the Fourier transform of the function.

Okay, so because this is true, and because f is a linear operator, so by linearity of f,