# Multi-electron eigenfunction problem

1. Apr 16, 2012

### RIPCLB

1. The problem statement, all variables and given/known data
"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

"

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

3. 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.

2. Apr 17, 2012

### 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.

Last edited: Apr 17, 2012