I was going through a QM proof which involved simplifying the following integral $$\int\int\left\langle\left.\left. x + \frac{\xi}{2} \right| \hat{\rho}\right|x - \frac{\xi}{2}\right\rangle \left\langle\left.\left. x - \frac{\xi}{2} \right| \hat{A}\right|x + \frac{\xi}{2}\right\rangle dxd\xi$$ (the $\langle\cdot|\cdot|\cdot\rangle$ notation refers to the standard Bra-ket notation and the bounds should be from $-\infty$ to $\infty$, but I don't think that's too relevant for my question). In particular, in order to prove what needs to be proved, I need to do a double integral change of variables with $u = x + \xi/2$ and $v = x - \xi/2$, but I realized I have no idea how to do this and that I've never done this before (I am neither a mathematician nor a physicist).
At first, I tried simply applying the definition of the total differential for both variables : $$du = \frac{\partial u}{\partial \xi}d\xi + \frac{\partial u}{\partial x}dx~\text{and}~dv = \frac{\partial v}{\partial \xi}d\xi + \frac{\partial v}{\partial x}dx$$ so that I get $$du = \frac{d\xi}{2} + dx~\text{and}~dv = -\frac{d\xi}{2} + dx$$ so that I can substitute expressions for $du$ and $dv$ and multiply them together: $$dvdu=\left(-\frac{d\xi}{2} + dx\right)\left(\frac{d\xi}{2} + dx\right)=d^{2}x-d\xi$$.
However, this is clearly wrong; I know I should get $dxd\xi=dvdu$, so that the integral simplifies to $$\int\int\left\langle\left.\left.u \right| \hat{\rho}\right|v\right\rangle \left\langle\left.\left. v \right| \hat{A}\right|u\right\rangle dvdu$$
I then googled some bit, and it seems you have to use the so-called Jacobian determinant in this substitution. In this case: $$dxd\xi= \begin{vmatrix}\frac{\partial x}{\partial v}&\frac{\partial x}{\partial u}\\\frac{\partial \xi}{\partial v}&\frac{\partial \xi}{\partial u}\end{vmatrix}dvdu$$ with $x = \frac{u + v}{2}$ and $\xi =u-v$, I then get $$dxd\xi= \begin{vmatrix}\frac{1}{2}&\frac{1}{2}\\-1&1\end{vmatrix}dvdu=1\cdot dvdu=dvdu$$ which is consistent with the answer, which I should be getting. However, I feel like I just conveniently defined my new variables so that this works out; I could have easily defined u and v the other way around, and then I would have gotten: $$dxd\xi= \begin{vmatrix}\frac{1}{2}&\frac{1}{2}\\1&-1\end{vmatrix}dvdu=-1\cdot dvdu=-dvdu$$ Said differently, I don't understand how to unambigously choose the order of my rows/columns in the Jacobian determinant, which would then change the sign of my final value (each time I permute adjacent rows/columns). How do I know which is the "first" old variable, the "second" old variable, the "first" new variable, etc.?
To summarize:
- Why is my approach with substituting the total differentials wrong?
- How do I unambiguously choose the order of my rows/columns in my Jacobian matrix (i.e. how do I know the final sign is correct)?
- In this particular case, is there maybe an easier way of substituting the variables (for such a simple substitution using the Jacobian determinant almost seems overkill).
Again, I come from a mathematically weak background, so I would maybe appreciate a more "intuitive"/less mathematically formal answer, if possible (for my first question, maybe there's a geometric way of looking at it as to why it doesn't work?).
