I'm applying a change of variables to the Schrodinger Eq, which means that my partial derivatives have gotten rather wonky.
In short: I've created new variables $\alpha \equiv x_1+x_2$ and $\gamma \equiv x_1-x_2$ to use with my SE since my potential function is in terms of those quantities. However, my partials are now $\frac{\partial^2}{\partial(\frac{\alpha+\gamma}{2})^2}$ and $\frac{\partial^2}{\partial(\frac{\alpha-\gamma}{2})^2}$. I cannot recall or devise a way to separate these partial derivatives into partials of purely $\alpha$ and $\gamma$. The Internet and old textbooks have been of no use here, so I think I'm either missing something trivial or this is much harder than it looks. I definitely want to keep using $\alpha$ and $\gamma$ since my potential function is now orders of magnitude simpler, and simplifying the partials will allow me to fully separate the equation and make the following math much, much easier.