How to calculate the transition rate

[Nastenka]

Homework Statement

I have the usual equation for the transition amplitude:
A=< k|exp(-iHt)| j>,while my Hamiltonian in Dirac notation looks like:

H=\sum E_a|a><a|+\sum_b (E_b|b><b|+V_1|0><b|+ V_2|b><0|)

In order to find the transition rate I should take a derivative as:
dA/dt, so that I will get something like:

A=< k|-iH*exp(-iHt)| j>
2. Relevant questions

Now, my question is:

how to treat it further?

The Attempt at a Solution

I know I should expand it with series for the exponent, but then I obtain in the middle this:
-iH(1+H)=-iH-iH² that confuses me. I feel I am stucked in such an easy task. But the proble is that I cannot expand it with eigenvalues and eigenvectors which would simplify a lot my task...

 

[mark726]

Hello,

Thank you for reaching out with your question. It seems like you are on the right track in expanding the exponent in series. As you mentioned, this will simplify the task by allowing you to use the eigenvalues and eigenvectors of the Hamiltonian.

To continue, you can use the following identities:

1. exp(A+B) = exp(A)exp(B) (assuming A and B commute)
2. exp(A) = I + A + A^2/2! + A^3/3! + ...

Using these identities, you can expand the exponent in your transition amplitude equation and then use the eigenvalues and eigenvectors to simplify the terms. This will allow you to take the derivative with respect to time and obtain the transition rate.

I hope this helps. Let me know if you have any further questions or if you need clarification on any of the steps. Good luck with your calculations!