# Anisotropic harmonic oscillator

by dingo_d
Tags: anisotropic, harmonic, oscillator
 P: 211 1. The problem statement, all variables and given/known data 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. 3. 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?