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I am looking at the coupling hamiltonian for electrons in an EM field. In particular I'm interested in the inelastic scattering (this isn't the dominant part for inelastic scattering but it's confusing me).

The part of the hamiltonian in the time/space domain that I'm interested in is

[itex]H_A = (\frac{e^2}{2mc^2})\sum_j A(r_j,t) \cdot A(r_j,t)[/itex]

Where [itex]j[/itex] sum across all the electrons in this many particle problem. [itex]A[/itex] is the vector potential of EM field.

Now I have another paper which fourier transforms this part of the Hamiltonian as

[itex]H_A = (\frac{e^2}{2mc^2})\sum_{k_1,\omega_1} \sum_{k_2,\omega_2} N(-k_1 + k_2) A(k_1,\omega_1) \cdot A^*(k_2,\omega_2)[/itex]

where

[itex] N(-k_1 + k_2) = \Sigma_j e^{i(k_1 - k_2)} \cdot r_j [/itex]

which is the fourier transform of the many particle number operator for the electrons.

The question

How exactly is the hamilton fourier transformed in this way?

It seems to imply that

[itex] A(k_1,w_1) = \Sigma_{k_2,\omega_2} N(-k_1 + k_2) A^*(k_2,\omega_2) [/itex]

assuming that the fourier transformed hamiltonian can be written as

[itex]H_A = (\frac{e^2}{2mc^2})\sum_{k_1,\omega_1} A(k_1,\omega_1) \cdot (k_1,\omega_1)[/itex]

which I am not certain of..

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# Fourier transform of scattering hamiltonian

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