# Expectation values of harmonic oscillator in general state

1. Nov 19, 2012

### Jomenvisst

So, this has been bothering me for a while.

Lets say we have the wavefunction of a harmonic oscillator as a general superposition of energy eigenstates:

$\Psi = \sum c_{n} \psi _{n} exp(i(E_{n}-E_{m})t/h)$

Is it true in this case that <V> =(1/2) <E> .

I tried calculating this but i get something like

<V> = < $\Psi |V| \Psi$ > = (1/2)<E> + some other term that does not seem to be zero generally.

However, it seems to me that <V> =(1/2) <E> should be true even in this case, since
$<V>_{n} = <\psi_{n} | V | \psi_{n} > = (1/2) <E>$ for every n.

2. Nov 19, 2012

### dextercioby

The superposition state is no longer stationary. The derivation of the virial theorem (Davydov, Sect 17 from Chapter II) will not be valid.

3. Nov 19, 2012