Wave packets that feel harmonic potential

[jostpuur]

Are there any nice wave packets you could write as a superposition of eigenstate solutions of a one-dimensional harmonic oscillator? The question deals with a situation, where a particle feels a harmonic potential, but is far away from the center and is traveling as a wave packet, probably oscillating like a classical particle before spreading.

I tried the usual gaussian wave packet, but it lead to an integral

$$\int\limits_{-\infty}^{\infty} H_n(x) e^{-Ax^2+Bx}dx$$

which I found too difficult for myself. Do you know if a foolproof technique already exists for integrating this, or if there is other kind of wave packets that are easier?

$$H_n$$ is Hermite's polynomial.

 

[variation]

Hello,

The problem you meet seems "how to do the integral $$\int_{-\infty}^{+\infty}H_n(x)e^{-Ax^2+Bx}dx$$"
I try your problem as far as i can ( with brute force >"< ):
In the begining, use the generating function of Hermite polynomials
$$e^{-t^2+2tx}=\sum_{n=0}^{\infty}\frac{1}{n!}t^nH_n(x)$$
Multiply $$e^{-Ax^2+Bx}$$ both sides and integral over all $$x$$:
$$\text{L.H.S.}=\int_{-\infty}^{+\infty}e^{-t^2+2tx}e^{-Ax^2+Bx}dx=\sqrt{\frac{\pi}{A}}e^{-t^2}e^{\frac{(B+2t)^2}{4A}}$$
$$\text{R.H.S.}=\sum_{n=0}^{\infty}\frac{1}{n!}t^n\int_{-\infty}^{+\infty}H_n(x)e^{-Ax^2+Bx}dx$$
One can calculate the Taylor expansion of the L.H.S. and find the coefficient of the $$\frac{t^n}{n!}$$ term:
$$\sqrt{\frac{\pi}{A}}\left(\frac{\partial^n}{\partial t^n}e^{-t^2}e^{\frac{(B+2t)^2}{4A}}\right)_{t=0}$$
The integral you want appeals in the Taylor expansion coefficient of $$\frac{t^n}{n!}$$ term:
$$\int_{-\infty}^{+\infty}H_n(x)e^{-Ax^2+Bx}dx=\sqrt{\frac{\pi}{A}}\left(\frac{\partial^n}{\partial t^n}e^{-t^2}e^{\frac{(B+2t)^2}{4A}}\right)_{t=0}$$
Finally, it can be calculated further with Leibniz rule and obtain the result:
$$\int_{-\infty}^{+\infty}H_n(x)e^{-Ax^2+Bx}dx=\sqrt{\frac{\pi}{A}}e^{\frac{B^2}{4A}}\sum_{N=0}^{[\frac{n}{2}]}\frac{n!}{(n-2N)!N!}\left(\frac{1}{A}-1\right)^N\left(\frac{B}{A}\right)^{n-2N}$$
where $$[x]$$ gives the maximum integer that is equal to or less than $$x$$.


Best regards

 

[jtbell]

I'm on the road right now and don't have access to my books, but I do seem to remember that it's possible to construct a Gaussian wave packet for the SHO, and that in fact it doesn't "spread." That is, it's width $\Delta x = \sqrt {<x^2> - <x>^2}$ remains constant as the packet moves back and forth. I remember doing this as an exercise in grad school.

 

[jostpuur]

Nice trickery, variation! I just tried integration by parts and recursion relations of Hermites polynomials, without success. Seems I should add more tricks to my arsenal.

 

[Hans de Vries]

Here you have them numerically:

the non-spreading version:

http://chip-architect.com/physics/gaussian.avi

A narrow Gaussian cyclically spreading out and contracting back:

http://chip-architect.com/physics/narrow_gaussian.aviI wanted to visualize both the phase as well as the magnitude. It shows
nicely how the phase change on the x-axis corresponds to the local
momentum. The one spreading out has momentum going both ways
while spreading out and contracting back again.
Regards, Hans

 

[Mentz114]

Jostpuur, this is nicely done in 'Quantum Theory' by David Bohm (1951) Dover.
On page 307 he derives the wave packet for the QSHO and gets the non-spreading WP.