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So hello and welcome to lecture number nine.

Today we'll take the machinery we developed so far and apply it in what I call a case study

because in fact this is an application of the machinery developed so far

that will show all the subtleties we talked about

and we'll talk about a very important object in quantum mechanics, the so-called momentum operator.

So we put the machinery to work and we will develop a little more machinery

but it's good to illustrate what we did so far abstractly in a concrete example.

We consider the so-called momentum operator

and this momentum operator is an object we're going to call P, capital P, because it's the operator.

It will have a domain which we call curly D P

and it will map in the full Hilbert space of which this is to be a dense subset

and the complete Hilbert space is L2 of R D

but in our example here we will specialize to D equals one

because that shows the principle well enough but in general it's from D P to L2 R D

and what is this operator doing?

It takes an element of L2 R D or rather a suitable dense subset thereof

and maps it to minus i h bar del J

so that would be the Jth momentum operator

that's the component of the momentum, the Jth component and it acts on psi.

Okay so this is what you will see in the quantum mechanics books as the momentum operator.

Now the question is why is this the momentum operator?

Why does it look roughly like this?

The answer is provided by a very deep theorem, is the Stone-Von Neumann theorem

that says that up to unitary equivalence this is unique.

It says this is up to unitary equivalence

the unique momentum operator in d spatial dimensions.

So now I'm not talking about the dimensions of the Hilbert space

but the dimensions of the physical space, think of classical mechanics,

in which the particle moves.

So if D equals one we have a particle that moves only in one dimension.

This will come later because in order to even write down the assumptions of the Stone-Von Neumann theorem

we need the spectral theorem first.

Okay so this why, please take it on trust right now

that this is an important operator in quantum mechanics

because it corresponds to the momentum observable in quantum mechanics.

However there is still an open question at this point

and that is how to choose the domain of this operator.

Because fair enough you see if you take square integrable functions

not every square integrable function can be differentiated.

A square integrable function could have a jump in one point, could be discontinuous.

Well if it's discontinuous it cannot be derived.

Hence we must restrict this operator.

Moreover once we will be able to formulate the canonical commutation relations

we will see that we always need to go to an infinite dimensional Hilbert space

because this is an unbounded operator and so on.

But right now what we do, we'll play the following game.

We'll play the game that you will be confronted with over and over again.

People will say let's take this as the momentum operator

and they will not specify the domain on which it acts.