Okay, hello. We are just studying geometric phases and so last time we discussed one example

which was the R1 of boom phase. So it's called a geometric phase because it's a phase shift

that only depends on the path, not on the velocity with which you traverse the path.

But on top of that it turns out the R1 of boom phase was then even a topological phase

because it could even distort the path and as long as the distortion doesn't take you

across any area where there is a magnetic field then the phase is just constant.

And so now we are going to generalize this and really treat true geometric phases where

things do depend on the path and these are usually nowadays referred to as belly phases.

The idea is that somewhat in contrast to what we discussed for the R1 of boom effect, we

imagine adiabatic variations of a parameter in an Hamiltonian in such a way that we start

in an eigenstate of the Hamiltonian and we always remain in an instantaneous eigenstate

dragging along a wave packet for example and then trying to figure out what happens if

we perform this time evolution such that we return to the starting point. We change parameters

adiabatically, we return to the starting point, principle nothing has changed but of course

you have accumulated a phase. What is this phase? And it will have two components. One

is the simple energy times time dynamical phase as it is called but there is another

part and that turns out to be a geometric phase.

Okay, so this was discussed by Michael Berry in 1984 but as is so often the case with these

things, it had actually been discussed earlier by an Indian physicist, Anshar Ratnam, already

back in 1956 but apparently since not everyone reads the proceedings of an Indian journal,

the name Berry got attached to it. Now the setting is very simple. Imagine for example

we have some potential and in this potential we have trapped a particle and say it is in

the ground state of this potential and now we are starting to slowly move around the

potential in practice that could work for example if you change the voltages on the

electrodes that control the motion of some electrons inside a semiconductor. What could

work if you consider atoms trapped in an optical dipole potential and you change the arrangement

of the lasers. In any of these cases you can just move around the potential. So I will

say that potential depends on position but also it depends on some parameter which for

example would be the location of the minimum of the potential and in turn this parameter

that is called a lambda also depends on time. So that is the setting and so we are moving

around the position of this potential minimum and going somewhere else and we are doing

it so slowly that all of the time the wave function, the quantum state basically remains

in the ground state by whatever energy eigenstate you start with. So that is the idea. Now instead

of changing the potential you could also change the mass if you can and so in general I will

just assume that we have a Hamiltonian that is really a function of a parameter that depends

on time. Now you could say why don't I just write the Hamiltonian depending on time which

would be sort of equivalent but the idea is that in the end I want to imagine many different

situations where the parameter trajectory is actually the same but it's traversed with

different speed and then we will see there is one part to the face that does not depend

on the speed and there is another part that really does depend on the time evolution.

So our goal in principle is simply to solve the Schrodinger equation. So we want to solve

the time dependent Schrodinger equation which looks like this. Now as I said the idea is

that if you change things very slowly the particle will remain in the instantaneous

eigenstate so that will be our answer and there will be small corrections to that of

course because we will never move infinitely slowly so there will be a slight excitation

to higher energy eigenstates but this we will be able to neglect for our purposes. So what's

our ansatz? We say psi of t is given by the instantaneous eigenstate up to a phase vector

and so the instantaneous eigenstate I will call psi 0 that only depends on lambda and

of course lambda depends on time plus there will be this phase vector in front that I

call e to the i phi of t and then there will be corrections that depend on the speed with