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Anisotropic harmonic oscillator 
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#1
Jan1011, 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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