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 .
Example Suppose a particle is described by the wave function
where 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