When differential equation like Schrodinger is separable in some coordinate system?

Question

When differential equation like Schrodinger is separable in some coordinate system? What needs to satisfy the potential ?

Answer 1

I'm not entirely sure what you are trying to ask, but I think it is this: When is the Schrodinger equation (or a similar differential equation) separable? What conditions must the potential function satisfy?

The short answer is that the Schrodinger equation is separable when the potential is independent of time (though there maybe time independent potentials that also work).

A differential equation of two independent variables is separable if the equation can be algebraically manipulated such that only one type of variable appears on each side of the equation. In the case of a partial differential equation (i.e. the Schrodinger equation) dependent variable can be written as a product of functions of the two independent variables; that is

$$ \Psi(x,t) = \rho(x)\phi(t) $$

If we apply the Schrodinger equation to this "guess" and assume $V$ is independent of time we find (after a few steps):

$$ -\frac{\hbar^2}{2m} \frac{d^2\rho}{dx^2} = (E-V)\rho $$

for and $E=$ constant.

Note that this equation is an ordinary differential equation though we started with a partial differential equation. More importantly, since $\rho(x)$ is independent of time, and therefore so is this entire equation. Hence, it is called the time independent Schrodinger equation.

This is essentially a shortened version of the derivation provided in Griffith's book.

Answer 2

Looks like the answer is given in the following article: L. P. Eisenhart, "Enumeration of potentials for which one-particle Schroedinger equations are separable", Phys. Rev. 74, 87-89 (1948). The list is too long to give it here.