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So good morning, welcome to lecture 12. Today we're going to consider periodic potentials.

In our systematic study of different types of potentials that does not restrict to very specific examples,

but rather than two classes of examples in order to have more general conclusions.

And periodic potentials means that we consider a Schrödinger operator H,

which is given by minus H bar squared over 2m as always,

with some Laplacian plus a potential V that has the properties that A, this V is periodic.

That means if you shift, so let me explain this in one dimension,

but you can easily have a, say, three-dimensional R3 element.

If you have A in R3, then you have a similar thing, you can have several such.

But let's restrict to one dimension for simplicity,

such that you for all x in R3, you have this property.

So it extends infinitely far, but it's periodic.

And there's a second technical condition we need to impose.

Without that condition, we cannot make far-reaching conclusions,

and that is that the potential V is piecewise continuous, piecewise continuous and bounded.

So we don't want unbounded potentials and bounded.

So piecewise continuous, of course, means that,

so if you draw a graph of the potential V of x over x,

again, restricting to a one-dimensional model,

it just means that whatever the potential looks like, piecewise continuous and bounded,

ta-da-da-da, then it can make a jump, say something like this,

and then it can do something else.

But from then on, you need to repeat precisely that.

So from then on, say, it again goes like...

and so on.

And obviously the A we're talking about is then precisely from here to here.

This is the A.

And the potential is bounded, so we have constants above and below,

such that it looks somewhat like this.

So imagine whatever potential of such type you come up with.

We want to extract physics from that without knowing precisely,

without knowing precisely how this potential looks like.

And at first sight, if you look at a problem like this...

At first sight, if you look at a problem like this,

it seems particularly hopeless, because this Hamiltonian, the Schrodinger operator,

severely depends on the potential, and the assumption,

the hope that you can derive anything in general about a situation like this,

I think, is a priori very low.

But in fact we can, and this has important applications.

So fact, one can, well, and we will today,

extract a remarkable generic conclusion

about the energy spectrum of a particle moving in such potentials, in any such potential.

I will tell you before we start what the conclusion is,

so you know what we're aiming for.

But the application of this is maybe one of the most heavily used applications of physics today.

It's in solid state physics.

So in solid state physics, a very simple model of a solid body,

as seen from the perspective of an electron that hops around on it,

is you can have irregular lattice.