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Say I'm given the following Schrodinger equation

$$\frac{d^2u}{dx^2}+ \left[E - V(x)+ \frac{a}{x^2}\right]u(x) =0$$

Where $a \in \mathbb{R}$. What are the physical interpretations of this equation? I understand that it means the angular momentum is fixed, but are there any other implications associated with this?

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This kind of reminds me of the centrifugal Schrödinger: en.m.wikipedia.org/wiki/…. Sorry whiskey might affect. –  Shut up and Calculate Jan 26 '14 at 1:49
Go further down in the link to radial... You will see a 1/r^2 term. –  Shut up and Calculate Jan 26 '14 at 1:52

1 Answer 1

Given the constraint that $a$ be any real number, this equation can only be interpreted as the Schrödinger equation for a massive non-relativistic particle in one dimension, subject to the potential $V(x)-a/x^2$. If $V(x)$ is sufficiently regular at the origin, the equation will have a fairly ugly singularity there.

The equation does, as you note look superficially like the Schrödinger equation of a particle in three dimensions, subject to a central potential which preserves angular momentum, once the angular dependence has been separated. However, in that case the possible values of $a$ are much more restricted.

This is because the singular term $a/r^2$ and the second derivative both come from the kinetic energy term described by the laplacian, $\nabla^2$, and this means that the same constants that multiply the radial derivative must also multiply the kinetic energy term. In such an equation, one only ever encounters the combination $$ \frac{d^2}{dr^2}-\frac{l(l+1)}{r^2}, $$ where $l=0,1,2,\ldots$.

The reason for this is that when you do a separation of variables into a radial and angular part as $\Psi(r,\theta,\phi)=\frac1r u(r)Y(\theta,\phi)$, the angular part of the wavefunction is constrained to obey an eigenvalue problem of its own, $$ \frac{1}{\sin^2\theta}\left[ \sin\theta\frac\partial{\partial\theta} \sin\theta\frac\partial{\partial\theta} +\frac{\partial^2}{\partial\phi^2} \right] Y(\theta,\phi)=aY(\theta,\phi). $$ The solutions of this eigenvalue problem are quite 'rigid', and the only way to have regular solutions is, as usual, for the constant $a$ to be restricted to a specific set. Here, it must obey $a=-l(l+1)$ for $l$ a nonnegative integer.

Thus, if your constant does obey such a condition, then yes, your equation does admit interpretation as the radial equation of a 3D particle with conserved angular momentum, and the singular term represents the centrifugal barrier. Otherwise, no: it can only be interpreted as a 1D Schrödinger equation.

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nice answer, i completely agree. +1 –  Zoltan Zimboras Jan 26 '14 at 3:07
I've been informed that in 2-dimensional systems it is possible for the angular momentum to take on any possible value, and it is related to the physics of anyons.. see the following paper susyqm.com/wp-content/uploads/2012/11/… (page 2) for more information. –  user119264 Jan 26 '14 at 4:17

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