Time Evolution of Gaussian Wave Packet in Anisotropic Harmonic Oscillator

[dingo_d]

Homework Statement

So this kinda incorporates my last questions. I have a particle described by a Gaussian wave packet. And it moves in 2D anisotropic h.o potential with commensurate frequencies (1:2). I've solved the x part (I was messing around with the nasty integral, which in the end turned out to be easy, because I chose my wave function at t=0 with wrong constants).

And I turned to y part.

The initial wave function is:

$$\psi(y,0)=\left(\frac{m\omega_y}{\pi\hbar}\right)^{1/4}e^{-\frac{\lambda y^2}{2}}$$

where:
$$\lambda=\frac{m\omega_y}{\hbar}$$.

I need to find out the time evolution of the system by expanding it in the basis of eigenstates of anisotropic harmonic oscillator (and since it's just the multiple of 1D harmonic osc. I just use the Y part here).

So:

since $$\psi(y,0)=\sum_{n_y=0}^\infty c_{n_y}(0)\psi_{n_y}(y,t)$$ and

$$c_{n_y}(0)=\int_{-\infty}^\infty \psi(y,0)Y_{n_y}^*(y)$$

My $$c_{n_y}(0)$$ is (with variable change $$\sqrt{\lambda}y=\eta$$):

$$c_{n_y}(0)=\frac{1}{\sqrt{\pi}}\frac{1}{\sqrt{2^{n_y}n_y!}}\int_{-\infty}^\infty e^{-\eta^2}H_{n_y}(\eta)d\eta$$

Mathematica and sources on the internet says:$$\int_{-\infty}^\infty e^{-\eta^2}H_{n_y}(\eta)d\eta=-\frac{\sqrt{\pi}2^{n_y-1}\left[(n_y-2)_2F_1\left(1;\frac{1-n_y}{2};\frac{1}{2};1\right)+1\right]}{\Gamma\left(1-\frac{n_y}{2}\right)}$$

And that is 0 :\

So what am I doing wrong?

The reasoning is that, because there is no offset in y direction of the Gaussian the time evolution will be that the function never changes with time (the probability density will be static).

I solved x part and I got that, as my time goes by, the x part (probability) is oscillating (depending on the initial displacement and other constants). And as I make initial displacement smaller it just stays still for t>0.

But the problem is, if my coefficient in t=0 is zero and coeff. at t is given by:

$$c_{n_y}(t)=c_{n_y}(0)e^{-\frac{\imath}{\hbar}E_{n_y}t}$$ all coefficients will be zero! and then my wave function given by:

$$\psi(y,t)=\sum_{n_y=0}^\infty c_{n_y}(t)Y_{n_y}(y)$$ will be zero :\

Or am I missing sth here?

 

[NateD]

Homework Equations \psi(y,0)=\left(\frac{m\omega_y}{\pi\hbar}\right)^{1/4}e^{-\frac{\lambda y^2}{2}}c_{n_y}(t)=c_{n_y}(0)e^{-\frac{\imath}{\hbar}E_{n_y}t}\int_{-\infty}^\infty e^{-\eta^2}H_{n_y}(\eta)d\eta=-\frac{\sqrt{\pi}2^{n_y-1}\left[(n_y-2)_2F_1\left(1;\frac{1-n_y}{2};\frac{1}{2};1\right)+1\right]}{\Gamma\left(1-\frac{n_y}{2}\right)}The Attempt at a SolutionI think I'm missing sth here. Because this integral is supposed to be zero, but then my time evolution coefficients will be zero, which doesn't make sense.