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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:…. Sorry whiskey might affect. – Faq Jan 26 '14 at 1:49
Go further down in the link to radial... You will see a 1/r^2 term. – Faq Jan 26 '14 at 1:52

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… (page 2) for more information. – user119264 Jan 26 '14 at 4:17

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