I'm considering the time-independent Schrödinger equation in two dimensions,
$$\frac{-\hbar^2}{2m}\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right)\psi + U(x,y)\,\psi = E\,\psi \ \ .$$
Textbooks usually consider the case of a constant or zero potential $U$ within some boundaries. The way to solve the equation would then be to separate the variables, $\psi(x,y) = f(x)\cdot g(y)$.
$U$ being constant allows to separate the equation, e.g. putting all $x$ dependence on one side of the equation and all $y$ dependence on the other side.
Now, I am interested in the case where $U(x,y)$ is not (piecewise) constant, and also not only dependent on one single variable.
It seems to me that in this case separation of the variables does not necessarily work anymore. However, in how far is this generally true? Are there classes of potentials, for which the Schrödinger equation still is separable?
Intuitively I thought that potentials like
- $U(x,y) = v(x) + w(y)$ or
- $U(x,y) = v(x)\cdot w(y)$
should still be somehow special in the sence that also the solution $\psi$ would be separable in one way or the other. For the case of additive separability (case 1.) of the potential (like the harmonic oscillator) this seems to be the case, while for the second case not, although they would share the same symmetry.
Is there some general law behind that, e.g. additively separable potentials give separable solutions, multiplicatively separable potentials don't? Is my intuition wrong?
Best Answer
The following article may be relevant: L. P. Eisenhart, "Enumeration of potentials for which one-particle Schroedinger equations are separable", Phys. Rev. 74, 87-89 (1948) I read it a few years ago, but I don't have immediate access to it right now. I remember it contained a long list of potentials where separation of variables is possible.