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

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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.

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This is not the question--- the question is about separation of spatial variables into one dimensional differential equations. This answer is more trivial than the question, since time separability is not a significant simplification. –  Ron Maimon Sep 23 '12 at 4:16
    
Oh I see. I haven't even begun doing QM in three dimensions yet. Thanks for explaining what the question was. I suppose "some coordinate system" should have made it fairly clear. So do all these solutions assume the potential to be independent of time? –  PatEugene Sep 23 '12 at 4:31
    
Hi @PatEugene, and welcome to Physics.SE. The mathjax formatting we use delimits TeX with single dollar signs (inline math) or double dollar signs (set-apart math). I've edited your post accordingly (subject to peer approval), and so you can see how the source looks (you can click edit without making any changes). –  Chris White Oct 2 '12 at 3:30
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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.

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