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

 

[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.