Time-Dependent Frequency Harmonic Oscillator

[canbula]

Homework Statement

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.

Homework 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
$V(t) = - \frac{1}{2} m \omega_{0}^2 \lambda t x^{2}$

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.

 

[Thaakisfox]

What are the relevant equations for time dependent perturbation theory?

 

[canbula]

sorry I added them to my original post

 

[Thaakisfox]

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

 

[paranormal]

I would approach this problem by first understanding the concept of time-dependent perturbation theory. This theory allows us to calculate the time evolution of a quantum system when there is a time-dependent perturbation present. In this case, the time-dependent perturbation is the changing frequency of the harmonic oscillator.

To find the time dependence of the ground state energy, we can start by using the Schrödinger equation to determine the time evolution of the ground state wavefunction. We can then use the time-dependent perturbation theory to calculate the energy shift caused by the time-dependent frequency.

Using the given harmonic oscillator Hamiltonian, we can write the Schrödinger equation as:

i\hbar \frac{\partial \Psi}{\partial t} = \left[ \frac{p^2}{2m} + \frac{1}{2} m \omega (t)^2 x^2 \right] \Psi

We can then expand the time-dependent frequency term using the given power series expansion:

\omega (t) = \omega_0 * \exp^{- \lambda t} = \omega_0 * (1 - \lambda t + \frac{\lambda^2 t^2}{2!} - ...)

Plugging this into the Schrödinger equation, we get:

i\hbar \frac{\partial \Psi}{\partial t} = \left[ \frac{p^2}{2m} + \frac{1}{2} m \omega_0^2 (1 - \lambda t + \frac{\lambda^2 t^2}{2!} - ...) x^2 \right] \Psi

Now, we can use the time-dependent perturbation theory to calculate the energy shift caused by the time-dependent frequency. The first-order energy shift is given by:

\Delta E^{(1)} = \langle \Psi_0 | V(t) | \Psi_0 \rangle

where \Psi_0 is the ground state wavefunction and V(t) is the time-dependent potential. Using the given potential, we get:

\Delta E^{(1)} = - \frac{1}{2} m \omega_0^2 \lambda t \langle \Psi_0 | x^2 | \Psi_0 \rangle

Since the ground state wavefunction is an even function, we can write:

\langle \Psi_0 | x