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$$ \Delta \Psi(x, y, z) + V(x, y, z)\Psi(x, y, z) = E \Psi(x, y, z) $$

For which $V(x, y, z)$ can this partial differential equation (eigenproblem) be solved by separation of variables?

Ystar
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1 Answers1

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Apply separation of variables

$$\Psi(x,y,z) = X(x)Y(y)Z(z)$$

and the equation becomes

$$\frac{d^2X}{Xdx^2} + \frac{d^2Y}{Ydy^2} + \frac{d^2Z}{Zdz^2} + V(x,y,z) = E$$

Now in order to be able to split this into three seperate equations we cannot have any coupling, i.e. cross-terms like $xy^2$, between any of the variables in $V$. Thus we must have

$$V(x,y,z) = A(x) + B(y) + C(z)$$

Disclaimer: For some equations you might be able to perform a change of variables $(x,y,z)\to(x',y',z')$ where we have cross-terms in $V(x,y,z)$ but not in $V(x',y',z')$. Now if the Laplacian in the new variables still remains simple in the sense that it does not mix derivatives of $(x',y',z')$ then seperation should also work here. How to know if this can be the case for a given equation (or if this is possible at all): I have no clue!

Winther
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    Some obscure math. physics rambling towards your last point: With regards to the Schrodinger equation in quantum mechanics (which this PDE is), the question of whether the potential is separable in some coordinates is linked with the integerability of the corresponding classical system, and to applicability of semiclassical methods in these cases . – Semiclassical Jul 20 '14 at 19:03
  • (Which is to say, it's quite a deep subject.) – Semiclassical Jul 20 '14 at 19:06
  • @Semiclassical Nice point. Though, way above my league:) – Winther Jul 20 '14 at 19:48
  • Above my own as well, frankly. Just wanted to point out that the distance between "is this problem separable in these coordinates" and "is it separable in some coordinates" is indeed a vast one and one in which a lot of work has been done. – Semiclassical Jul 20 '14 at 19:50