# Particle in a potential well of harmonic oscillator

1. May 19, 2013

### Roodles01

1. The problem statement, all variables and given/known data
I have a similar problem to this one on Physicsforum from a few years ago.

2. Relevant equations
Cleggy has finished part a) saying he gets the answer as
Ψ(x, t) = (1/√2) (ψ1(x)exp(-3iwt/2+ iψ3(x)exp(-7iwt/2)

OK
classical angular frequency ω0 = √C/m for period of oscillation T = 2∏ / ω0
I note that E0 = ½ ħ ω0

3. The attempt at a solution
I have, for a wave packet with equal coefficients for 1st & 2nd stationary state wave function;
ψA(x,t) = 1/√2 (ψ0(x) e-iwt/2 + ψ1(x) e-3iwt/2)

This question asks for vaidity for t>0

I don't get how he got there.

Last edited: May 19, 2013
2. May 19, 2013

### Thaakisfox

Suppose $$\psi_n(x,t)$$ are the eigenstates with energy $$E_n$$.

Then a generic state (general solution of the Schrodinger equation) can be written as the superposition of these states:

$$\psi(x,t)=\sum_{n}a_n\psi_n(x,t)=\sum_{n}a_n\psi_n(x)e^{-iE_n t/\hbar}$$

Now match this with the initial condition to find the coefficients a_n and use the energy states for the harmonic oscillator:

$$E_n=\hbar\omega\left(n+\frac12\right)$$

3. May 19, 2013