I am computing matrix elements of a two body quantum-mechanical potential, which take the form(adsbygoogle = window.adsbygoogle || []).push({});

[tex] V_{k l m n} = \int d^3 r_1 d^3 r_2 e^{-i k \cdot r_1} e^{-i l \cdot r_2} V( | r_1-r_2 | ) e^{i m \cdot r_1} e^{i n \cdot r_2} [/tex]

To do this integral, I make the change of coordinates

[tex] \overset{\rightarrow}{r} \equiv ( \overset{\rightarrow}{r}_1 - \overset{\rightarrow}{r}_2 ) / 2, \overset{\rightarrow}{R} \equiv ( \overset{\rightarrow}{r}_1 + \overset{\rightarrow}{r}_2 ) / 2 ,[/tex]

which gives a momentum conserving delta function times the Fourier transform of the potential. This is exactly what as expected, but I am concerned that I am missing an overall Jacobian factor when I make the swap

[tex] d^3 r_1 d^3 r_2 \rightarrow d^3 R d^3 r [/tex]

I know how to get Jacobians for a single particle's coordinates, but for some reason I can't think straight about two particles. Can anyone provide guidance on this issue?

thanks :)

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# Jacobian for CoM coordinates

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