Okay, so we are in the middle of discussing quantum dissipative systems and there is one

correction I want to explain because last time I wrote down the decay rates as calculated

from Fermi's golden rule and I found there was a factor of 2 pi a mistake which can happen

of course. So this is just a repetition when we have a transition between two levels and

the transition rate can be written in terms of the matrix element of the system operator

if we have an interaction between a system and the environment written like this system

operator multiplied with some fluctuating force. So it's a matrix element of a system

operator between the final and the initial state and there is a pre-factor of 1 over

h bar squared and here last time I had a 2 pi mistake indeed plus you have to evaluate

the spectrum of the fluctuations at a frequency which is given by the difference in the energies

between the initial and the final state. So in particular if the initial state has a higher

energy than the final state then you have to evaluate the spectrum at a positive frequency.

And so we deduced various things from that and let me just take this opportunity to explain

how this formula comes about because it's a nice little calculation. And the one remark

which we need for the derivation is simply the following. If I take something like this,

the expectation value of a product of fluctuating operator stegna two different types then this

is according to Heisenberg e to the plus i over h bar times the Hamiltonian that time

evolves my bath operator times f times e to the minus i over h bar. And so now the point

is that of course you can expand this in a complete set of phasor states, of energy eigenstates

for the bath. That is the first operator takes you from some initial bath state to some final

bath state then you time evolve this bath state so what happens here is you get e to

the minus i over h bar times the energy of the final bath state times t and then the

second f takes you back again to the initial bath state in order for this to be non-zero

and so the second e to the i ht will give you an e to the plus i over h bar and then

the energy of the initial bath state. So overall this will be a sum over all final bath states

taking the matrix element of f between the initial and final bath state and between the

final and the initial which is just a complex conjugate. So that is easy. And one has these

two oscillating factors. So now the idea is to take the Fourier transform of this and

we will see why. So you see that this exponential combines with those and all you get is a delta

function.

So

that is the Fourier transform of the correlator of the bath. And why is this

interesting for us? Well, if we apply Fermi's Golden Rule to the complete

problem initially of having a transition that takes simultaneously my system and

my bath from some old to some new state, then you know in principle how the

corresponding formula should look like. So Fermi's Golden Rule always tells you

take the matrix element of the corresponding perturbation, which in our

case would be the interaction part of the material, that is this one, and take it

between the initial and the final state. In this case it means initial and final

states of the combined system at a system plus bath. So the initial state

will consist of specifying both the initial state of the bath and the initial

state of my system and the final state likewise must be specified both with

regard to the bath and the system. So I take this matrix element squared, that's

one part of Fermi's Golden Rule, then I take an energy conserving delta function

that tells me if the perturbation is not time independent and that is a time

independent, if the perturbation is time independent and that is a time independent

perturbation, then the energies of the initial and final state must be the same.

And so the energies here are simply the sums of the energies of the system at

the bath.

So that is the energy conserving delta function and then as we are not