Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.6.5.
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In solving the Schrödinger equation for spherically symmetric potentials, we found that we could reduce the problem to the equation
where is related to the radial function by
We can write 1 as an eigenvalue equation for the operator in the form
We can show that, provided as , there are no degenerate eigenstates (that is, any state that is an eigenstate with energy is unique up to a scaling factor). The proof is similar to that in 1-d quantum mechanics, and goes by contradiction.
We suppose that there are two different functions and that satisfy 1 for the same energy (and same angular momentum number ). We then have
Multiply the first by and the second by and subtract to get
This expression is
which we can integrate to get
for some constant . This relation is valid for all , so we can choose where , which shows that . Therefore
Integrating gives us
for some other constant , so
That is, any two eigenfunctions with the same eigenvalue are multiples of each other, so represent the same state, which is nondegenerate.
Note that the derivation didn’t rely on the value of anywhere except at , so there is no requirement that, for example, as . Also, the derivation is valid whatever the sign of .