# Helps on understanding different representation transformations

1. Feb 25, 2012

### luxiaolei

Hi,all, I m an undergrades and I am suffering on understanding the different representation transformations, namely from schrodinger picture to interaction picture tupically, my lecturer didn't state which representation he was using and I m so confused, any helps would be great.

Shall I bring one example.

So I have an Hamiltonian for a 4 level ladder system, which consists a time independent and time dependent part(a perturbation from lasers),3 lasers which has frequency wa,wb,wc

H = H0+HI(t)

So, as far as I understood, this is not in schrodinger's picture. But confused it is in Heisenberg's or interaction pictures. So does the state vectors of above Hamiltonian, in which picture?

And if make a transformation in the way that

U= exp(iw1t)|1><1|+exp(i(w1+wa)t)|2><2|+exp(i(w1+wa+wb)t)|3><3|+exp(i(w1+wa+wb+wc)t)|4><4|

New state vectors = U*old state vectors
New Hamiltonian = U*H*U(dagger) - ihU*dU(dagger)/dt

So the new Hamiltonian becomes time independent , and state vectors becomes time dependent. So are they all now in the interaction pictures? Don't know why people do this transformation and what is the advantage?

Looks like to me this transformation gives a time independent Hamiltonian which is easier to work with, but what if I do want to keep the third laser time dependent, how should I construct this transformation?

I know, it's a lot questions, and I apologies for the lengthy newbie questions, and thanks so much in advance:)

Last edited: Feb 25, 2012