# Energies of a Quantum Harmonic Oscillator

Kyle91
Hey guys

I was just looking over a past homework problem and found something I'm not too sure on -

A particle is in the ground state of a Harmonic potential V (x) = 0.5mω2x2

If you measured the energy, what are the possible results, and with what
probabilities?

Now I know the answer to this is 0.5*hbar*ω and 100%. But I'm just a bit confused about when the formula for calculating this energy value can be applied.

E = 0.5*hbar*ω(n+0.5)

When can we use that? ^ Is it just for quantum harmonic oscillators or is it for all Quantum systems?

Cheers

## Answers and Replies

AlexChandler
If the state of the particle is the nth eigenstate, then you can use that last formula. When it is in the ground state, it still holds with n=0. When you are given some initial wave function that is not one of the eigenstates, you need to calculate the c values with fourier's trick, and then express the initial state as a linear combination of the eigenstates. Then the time dependent wave function is gotten by attaching the standard time dependence to each piece of the summation.