Jacobian for CoM coordinates

[UVCatastrophe]

I am computing matrix elements of a two body quantum-mechanical potential, which take the form

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

To do this integral, I make the change of coordinates
$$\overset{\rightarrow}{r} \equiv ( \overset{\rightarrow}{r}_1 - \overset{\rightarrow}{r}_2 ) / 2, \overset{\rightarrow}{R} \equiv ( \overset{\rightarrow}{r}_1 + \overset{\rightarrow}{r}_2 ) / 2 ,$$

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

$$d^3 r_1 d^3 r_2 \rightarrow d^3 R d^3 r$$

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 :)

 

[UVCatastrophe]

I figured it out...

Suppose $X \equiv (x + y)/2 , Y \equiv (x - y)/2$. Then $dx \wedge dy = (dX + dY) \wedge (dX - dY) = dY \wedge dX - dX \wedge dY = -2 dX \wedge dY$

I am being sloppy, because I am planning on throwing out that minus sign; perhaps someone can enlighten me (I think it has to do with orientation; in any case area elements can't be negative -- that would just be nonsense!)