# Harmonic Oscillator-Normalisation & Annihilation Operator

1. Apr 6, 2007

### n0_3sc

The problem statement, all variables and given/known data

Wavefunction:
$$\psi(x) = N\sum_{n=0}^\infty \frac{\beta^n}{\sqrt{n!}}\psi_n(x)$$
And $$\psi_n(x)$$ has eigenvalue $$E_n = (n + 1/2)\hbar\omega$$.

- Determine N (normalisation constant).
- Show $$\psi(x)$$ is an eigenstate of 'a' (annihilation operator).

The attempt at a solution

I don't know how to normalise it because $$\psi_n(x) \propto (a^+)^n \psi_0(x)$$ which makes things unusually complicated.

As for showing the eigenstate do I just operate 'a' on $$\psi(x)$$, expand, and get it in terms of $$\psi(x)$$ again?

(By the way how do I preview this post to check my tex? It keeps saying "Reload this page in a moment")

2. Apr 7, 2007

no worries.

All done :)