Discrete Spectrum Non-Degeneracy in 1D: How to Prove?

[andre220]

Homework Statement

Prove that in the 1D case all states corresponding to the discrete spectrum are non-degenerate.

Homework Equations

$$\hat{H}\psi_n=E_n\psi_n$$

The Attempt at a Solution

Okay so, what I am stuck on here is that the question is quite broad. I can think of specific cases like the 1D square-well where $E = \frac{n^2\pi^2\hbar^2}{2ma^2}$ which is non-degenerate. But in a more general sense this does not seem so easy to prove. We do know that the eigenvalues in this case are discrete ($E_n$) and the eigenfunctions are $\psi_n$, however I do not know where to go from here.

 

[BvU]

So basically what you want to prove is that if $\hat{H}(\psi_n-\psi_m)=0$, then ...