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Cross section computation - Huang's Statistical Mechanics

  1. May 19, 2012 #1
    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):

    [tex]T=<1',2'|T(E)|1,2>[/tex]

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

    [tex]T(E)=H'+H'(E-H_0+i\epsilon)^{-1}H'+\dots[/tex]

    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?
     
  2. jcsd
  3. May 19, 2012 #2
    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.
     
  4. May 19, 2012 #3
    Thank you very much for your reference! By pure coincidence I am also reading Sakurai's book, so I will definitely have a look.
     
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