# Quantum Harmonic Oscillator

1. Jan 25, 2009

### joker_900

1. The problem statement, all variables and given/known data
In the interval (t+dt, t) the Hamiltonian H of some system varies in such a way that |H|psi>| remains finite. Show that under these circumstances |psi> is a continuous function of time.

A harmonic oscillator with frequency w is in its ground state when the stiffness of the spring is instantaneously reduced by a factor (f^4)<1 such that its natural frequency becomes (f^2)w. What is the probability that the oscillator is subsequently found to have energy 1.5.(h/2pi).w.f^2?

2. Relevant equations
En = (n + 0.5)(h/2pi)w

3. The attempt at a solution

For the first part I thought maybe I needed to use ehrenfest, as |H|psi>|^2 = <psi|H|psi>. So then I think you get

d/dt <psi|H|psi> = <psi|dH/dt|psi>

However I don't really know what to do with this and am probably way off.

For the second part I'm really confused. The energy it talks about is the energy eigenvalue of the first excited state; but I don't know how to get a probability without the state |psi> being given to me. Initially the state is the ground stationary state for the original frequency w, so initially a measurement will definitely find the energy value (h/2pi)w. So the state must change, but how do I work out what it changes to?

Argh