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Imperatore
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Hey guys,
I signed up here because I needed some information on some quantum physics problems.
My question is related to quantum physics, and more precisely the derivation of time dependent perturbation theory. First of all, I am not able to understand all the maths structures and formulas, and it is in my opinion the biggest difficult I need to overcome to fully understand it. I checked some books and scripts just to find some more information about it, but unfortunately my doubts weren't stirred up. I hope that here I could find some more help and I really appreciate it.
So to begin with:
We get a hamiltonian, that can be written as
[tex]H(t)=H _{0}+V(t) [/tex], where [tex]V(t)[/tex] is a small time-dependent perturbation.
Let [tex]\psi ^{(0)} _{k} [/tex] be the wave functions of stationary states of unperturbed system with time multiplier. Any solution to the unperturbed wave equation can be expressed as the sum :
[tex]\psi= \sum_{k}a _{k} \psi ^{(0)} _{k}[/tex]
The solutions to perturbed equation:
[tex]ih \frac{\delta}{\delta t}\psi=(H _{0}+V)\psi[/tex] (1) (where h is Dirac constant, Planck constant divided by 2 Pi)
are in the form of:
[tex]\psi= \sum_{k}a _{k}(t)\psi ^{(0)} _{k}[/tex] (2)
Substituting (2) to the (1) and knowing that functins [tex] \psi^{(0)}_{k}[/tex] satisfy and equation :
[tex]ih \frac{\delta}{\delta t }\psi _{k} ^{(0)}=H_{0}\psi _{k} ^{(0)}[/tex]
somehow we obtain: (here is the problem for me. I don't understand how they solve below mentioned equations step by step)
[tex]ih \sum_{k}\psi _{k} ^{(0)} \frac{d}{dt}a _{k}(t)= \sum_{k} a_{k}V\psi _{k} ^{(0)}[/tex]
and then multiplying this equation from left side by [tex]\psi _{m} ^{(0)}* [/tex] and integrating we get :
[tex]ih \frac{d}{dt}a _{m}(t) = \sum_{k}V _{mk}(t)a _{k}(t) [/tex]
where [tex]V _{mk}(t)=\int \psi _{m} ^{(0)*} V\psi _{k} ^{(0)}dq=V _{mk}e ^{i\omega _{mk}t}[/tex]
and [tex]\omega _{mk}= \frac{E _{m} ^{(0)}-E _{k} ^{(0)} }{h}[/tex]
I will be very thankful if someone could explain it to me. I want to mention, that the majority of this text come from Landau book "Quantum mechanics. Non relativistic theory". At the end I want to add that I am Polish, so my English may be not perfect, but I still hope it is enough communicative ;)
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