Anisotropic harmonic oscillator

[dingo_d]

Homework Statement

The particle with the mass m is in 2D potential:

$$V(r)=\frac{m}{2}(\omega_x^2x^2+\omega_y^2y^2),\quad \omega_x=2\omega_y$$,

and is described with wave package for which the following is valid: $$\langle x\rangle (0)=x_0,\ \langle y\rangle (0)=0,\ \langle p_x\rangle (0)=0\ \textrm{i}\ \langle p_y\rangle (0)=0$$.

Expand the initial wave function by eigenstates of the anisotropic harmonic oscillator, and determine the time evolution of the system.
Find the energy and the angular momentum as a functions dependent of time and compare them with initial values.

Find the expected values of position and impulse and check the Ehrenfest theorem.

The Attempt at a Solution

Now I solved the time independent Schroedinger equation, for that potential and I got the energy eigenvalues with and without constrains, and it agrees with the solutions I found on the web and in the science articles.

What I'm a bit puzzled are these initial conditions. I'm given expected values in t=0. But solving the Schroedinger eq. I got solutions that have the form: $$\Psi(r,t)=R(r)\varphi(t)$$, where $$\varphi(t)=\varphi(0)e^{-\frac{iE_n}{\hbar}t}$$.

So how do exactly do those values come into play?

Or should I start with assuming the general Gaussian wave package, and try to run it through Fourier integral?

Or do I just need to add $$-x_0$$ to the x part of the solution? Since it's just Harmonic oscillator I would add that to exponental term.

But I'm not sure. Any ideas?

 

[mark726]

Thank you for your post. It seems like you have made good progress in solving the time independent Schroedinger equation for the given potential and obtaining the energy eigenvalues. Now, in order to determine the time evolution of the system, you can use the initial conditions given to you. These initial conditions represent the expected values of position and momentum at t=0.

To solve for the time evolution of the system, you can use the time-dependent Schroedinger equation, which is given by:

iħ ∂Ψ/∂t = HΨ

where H is the Hamiltonian operator. In this case, the Hamiltonian operator would be the anisotropic harmonic oscillator potential.

To determine the time evolution, you can use the initial conditions to construct the initial wave function Ψ(r,t=0). Then, you can use the time-dependent Schroedinger equation to solve for the wave function at any other time t. This would give you the time evolution of the system.

To find the energy and angular momentum as functions of time, you can use the time-dependent wave function obtained from the previous step. The expectation values of energy and angular momentum can be calculated using the time-dependent wave function.

In order to check the Ehrenfest theorem, you can use the time-dependent wave function to calculate the expected values of position and momentum at different times. The Ehrenfest theorem states that the expected values of position and momentum should follow the classical equations of motion.

I hope this helps. If you have any further questions, please feel free to ask. Good luck with your calculations!