# Time-Dependent Frequency Harmonic Oscillator

1. Dec 24, 2011

### canbula

1. The problem statement, all variables and given/known data
Consider an harmonic oscillator with time-dependent frequency as:
$\omega (t)=\omega_0 * \exp^{- \lambda t}$
Find the time dependence of the ground state energy of this oscillator for $\lambda << 1$ situation.

2. Relevant equations
$H=H_{0} + V(t)$
$H_{0} = \frac{p^2}{2m} + \frac{1}{2} m \omega_{0}^{2} x^{2}$
and if we use the power series expansion for $\lambda << 1$ we get
$V(t) = - \frac{1}{2} m \omega_{0}^2 \lambda t x^{2}$

3. The attempt at a solution
I know that I should use the time-dependent perturbation theory, but I am not good at it. So I need some help to solve this problem.

Last edited: Dec 24, 2011
2. Dec 24, 2011

### Thaakisfox

What are the relevant equations for time dependent perturbation theory?

3. Dec 24, 2011

### canbula

sorry I added them to my original post

4. Dec 25, 2011

### Thaakisfox

next, how do we express the energy eigenvalues in time-dependent perturbation theory? ;)