Helps on understanding different representation transformations

[luxiaolei]

Hi,all, I m an undergrades and I am suffering on understanding the different representation transformations, namely from Schrödinger 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 Schrödinger'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:)

 

[azntoon]

The transformation you have done is indeed a representation transformation from the Schrödinger picture to the interaction picture. In the Schrödinger picture, the state vectors are time-independent and the Hamiltonian is time-dependent; in the interaction picture, the state vectors are time-dependent and the Hamiltonian is time-independent.The advantage of the transformation is that it allows us to treat the time dependence of the Hamiltonian more easily. This is because in the interaction picture, the Hamiltonian is time-independent so we don't need to consider how it changes with time.If you want to keep the third laser time dependent, then you will need to use a different transformation. One way to do this would be to use a transformation that involves the exponential of the sum of the three frequencies of the lasers (wa, wb, wc). This will give you a time-dependent Hamiltonian but the state vectors will still be time-independent.