Multi-electron eigenfunction problem

[RIPCLB]

Homework Statement

"Prove that any two different nondegenerate bound eigenfunctions \psi_j(x) and \psi_i(x) that are solutions to the time-independent Schroedinger equation for the same potential V(x) obey the orthogonality relation

\int_-∞^∞ \psi_j^*\psi_i(x)dx=0

"

Homework Equations

I believe I have to find equations for which both eigenfunctions are solutions?

The Attempt at a Solution

I'm lost on how to get the problem started. I cannot think of any eigenfunctions to use. I might be putting more thought to it than I need to though.

 

[collinsmark]

Yes, the key is that they are eigenfunctions. But I think you're supposed to prove the orthogonality in the general sense, rather than use specific examples of eigenfunctions.

Suppose that \psi_a(x) and \psi_b(x) are eigenstates of the operator O, with corresponding eigenvalues a and b respectively.

O \psi_a(x) = a \psi_a(x)
O \psi_b(x) = b \psi_b(x)

Now consider each of these cases:

\int_{-\infty}^{\infty} [O \psi_a(x)]^* \psi_b(x)dx = \ \ ?

\int_{-\infty}^{\infty} \psi^*_a(x) [O \psi_b(x)]dx = \ \ ?

I'll let you take it from there.