Changing variables in the Schrodinger equation

[AxiomOfChoice]

Suppose I have a Schrödinger equation for two interacting particles located at x and y; so, something like
$$\left( i \frac{\partial}{\partial t} + \frac{1}{2m_x} \frac{\partial^2}{\partial x^2} + \frac{1}{2m_y} \frac{\partial^2}{\partial y^2} + V(x-y) \right) \psi(x,y,t) = 0.$$
Now, I want to shift to center of mass coordinates R and r and write the Hamiltonian in terms of them. The resulting Schrödinger equation looks something like
$$\left( i \frac{\partial}{\partial t} + H(R,r) \right) \varphi(R,r,t) = 0.$$
Question: is it true that
$$\psi(x,y,t) = \varphi \left(\frac{m_x x + m_y y}{m_x + m_y},x-y,t\right)?$$
If so, why? I have been trying to prove this but have so far just been going around in circles it seems...

 

[DrClaude]

Can you write $x$ and $y$ in terms of $R$ and $r$, such that you have a one-to-one mapping between the two? If yes, then there is no difference between expressing a function in terms of $(x,y)$ or of $(R,r)$.