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So good morning, welcome back. So having finished the formal part of the course, we'll have

a rather informal Chapter 6 applications.

And the first application we're going to look at is quantum mechanics on curved space.

Quantum mechanics on curved space.

So in this section and the next few we're going to discuss, I would like to change the

character of the course a little bit from telling the truth or telling the existing

structure to discovering together how in formulae and expressions and methods and folklore in

physics that you all know how we recognize the structures we found because that's good

training in order to apply the whole stuff.

And well if we look at quantum mechanics, let's look at it in a rather naive way and

that's the way it's presented usually in introductory courses. So in quantum mechanics we discuss

wave functions and such a wave function psi is if we talk for instance about some position

space representation, we think about it as an element of some space L2 over some R dimensional

space Rd so in most physics applications of course R3 as the underlying physical space.

And this L2 space, well actually we should talk, well there's a number of issues here

and it's the question, so let's call this Roman L2 Rd and let's construct this L2.

So let's first have a script L2 of Rd and let's say that's the space of all psi which

are complex valued functions on Rd such that a certain integral converges and the integral

dx psi absolute value squared converges. Now this space is not this space because in this

space we only look at the equivalence classes of such guys where it's ensured that the two

functions only differ, so any two elements in here are to be identified in here if the

difference has a vanishing integral of this type. So this is one of the issues. The other

issue is that here you need a Lebesgue integral and the other issue is that here you need

a Lebesgue integral in order to make this really work rather than a Riemann integral

and so on. So there are lots of things at this point to be said. However this is far

not all the subtlety one needs and I only mention this because I want to disregard this.

We're not going to look into these subtleties but they should be mentioned once. So if we

now look at some position space representation or some representation in general we wish

to have operators. We need self-adjoint operators and they should act on this space L2 but it

will turn out they can't. So let's call them Q i supposed to act on L2 linearly and Pi

supposed to act on L2 linearly i equals 1 to d. So this is the first step. So this is

the second step. So this is the third step. So this is the fourth step. So this is the

fourth step. So linearly i equals 1 to d and they're supposed to be self-adjoint with

respect to an inner product which in a sense are already used here phi comma psi d dx

psi star of x phi of x. And they're supposed to be self-adjoint and satisfied. They're

also supposed to satisfy the commutation relations that the Q i Q j is 0, the Q i P j is delta

i j and the Pi P j again ought to be 0 and here you may insert h bar but let's say h

bar is 1. Okay so this is certainly the standard folklore of what you want to do. Now it's

also the standard folklore that you can represent this operator Q i on psi, then called position

space representation by just multiplying with the i-th coordinate here. It's this guy and

the Pi psi can be represented by minus i h bar which was set to 1 del i psi of x. Now

the problem with this folklore is that if you have a square integrable function and

you derive it, well can you derive any square integrable function? No you can't because

you could have a function that just consists of little boxes you see and that can't be

derived at every point. So clearly this is the wrong domain if this guy is supposed to

be represented like this. Now let's assume we have a square integrable function and we

multiply it by a coordinate it depends on. The result needn't be square integrable anymore.

It's also pretty obvious. So it's also clear that this here is the wrong target. So all

of these things are taken care of. So these issues, issues of domains and targets. You