Cross section computation - Huang's Statistical Mechanics

[QuArK21343]

I am reading chapter three of Huang's Statistical Mechanics and I have a problem with equation (3.22). Having discussed the derivation of the classical cross section for a scattering process, Huang moves on to the quantum version of it. He states that in quantum mechanics the fundamental quantity in a scattering process is the matrix T, that represents an operator T(E):

$$T=<1',2'|T(E)|1,2>$$

where $|1,2>$ is the initial state ket of the system of two particles and the primed ket is the final state ket. He also writes:

$$T(E)=H'+H'(E-H_0+i\epsilon)^{-1}H'+\dots$$

where H_0 is the unperturbed hamiltonian, H' the potential and epsilon goes to zero. I don't understand what he means by that. Can you explain what I am missing or give me some reference? Up to now, the first two chapters were really good and I enjoyed them, but given that I have only a basic knowledge of non-relativistic quantum mechanics (one dimensional problems, bra-ket notation and not much more), do you think is it useful to go on reading this book?

 

[kof9595995]

You need some perturbation theory for QM, but Huang's stat mech doesn't rely too much on perturbation theory, so shouldn't be a big problem. For your question, this is essentially a Born approximation, c.f. Sakurai, "Modern quantum mechanics, revised edition", chap 7.2, and 7.2.20 is exactly what you wrote, but you probably need to read from the beginning of the chapter.

 

[QuArK21343]

Thank you very much for your reference! By pure coincidence I am also reading Sakurai's book, so I will definitely have a look.