Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Chapter 12, Exercise 12.6.1.

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The Schrödinger equation in 3-d for a potential that depends only on is

The angular part of the operator on the LHS is essentially the angular momentum operator (times ):

, so we can write this as

Eigenfunctions in this equation satisfy

where the subscript refers to the energy and the angular momentum quantum numbers and . is a spherical harmonic and is the radial function which depends on the potential . The eigenvalues of are so 3 becomes

We’ve dropped the from since, for a spherically symmetric potential, the radial function is independent of .

ExampleSuppose a particle is described by the wave functionwhere and are constants. What can we deduce about the system?

First, since is independent of and we see from 2 that

so the eigenvalue is and the state has no angular momentum. From 3 we therefore have

Working out the derivatives, we have

Plugging this back into 8 and cancelling terms gives

If as we have, in this limit

The energy is constant at all values of so we can now find from 11