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Anisotropic harmonic oscillator

by dingo_d
Tags: anisotropic, harmonic, oscillator
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dingo_d
#1
Jan10-11, 02:57 PM
P: 211
1. The problem statement, all variables and given/known data
The particle with the mass m is in 2D potential:

[tex]V(r)=\frac{m}{2}(\omega_x^2x^2+\omega_y^2y^2),\quad \omega_x=2\omega_y[/tex],

and is described with wave package for which the following is valid: [tex]\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 [/tex].

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: [tex]\Psi(r,t)=R(r)\varphi(t)[/tex], where [tex]\varphi(t)=\varphi(0)e^{-\frac{iE_n}{\hbar}t}[/tex].

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 [tex]-x_0[/tex] 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?
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