- #1

- 211

- 0

## Homework Statement

I have a function in t=0 given by:

[tex]\psi(x,0)=\frac{1}{\pi^{1/4}\sqrt{\sigma}}e^{-\frac{(x-x_0)^2}{2\sigma^2}}[/tex],

and I have to decompose it in eigenstates of harmonic oscillator given by:

[tex]u_n(x)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\frac{1}{\sqrt{2^nn!}}H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right)e^{-\frac{1}{2}\frac{m\omega}{\hbar}x^2}[/tex]

## Homework Equations

[tex]\psi(x,0)=\sum_{n=0}^\infty c_n(0)u_n(x)[/tex]

I use the fact that eigenfunctions of h.o. are ortonormal, and I can find:

[tex]c_n(0)=\int_{-\infty}^\infty\psi(x,0)u_n^*(x)dx[/tex]

## The Attempt at a Solution

And here comes the nasty part! Evaluating that integral. Now I can really move all the constants out, they play no vital role in the evaluation of that integral.

The integral is:

[tex]c_n(0)=\int_{-\infty}^\infty e^{-\frac{(x-x_0)^2}{2\sigma^2}}e^{-\frac{1}{2}\frac{m\omega}{\hbar}x^2}H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right)dx[/tex]

I can use the substitution [tex]\zeta=\sqrt{\underbrace{\frac{m\omega}{\hbar}}_{\lambda}}x[/tex], I can then sort things out a bit and get

[tex]\frac{1}{\sqrt{\lambda}}\int_{-\infty}^\infty e^{-\frac{(\zeta-\zeta_0)^2}{2\lambda\sigma^2}}e^{-\frac{\zeta^2}{2}}H_n(\zeta)d\zeta[/tex]

Where I've used the substitution: [tex]\sqrt{\lambda}x_0=\zeta_0[/tex]

Now I've solved similar integral by using generating function of http://en.wikipedia.org/wiki/Hermite_polynomials#Generating_function", but when I use that trick here I get:

[tex]\frac{1}{\sqrt{\lambda}}\int_{-\infty}^\infty e^{-\frac{(\zeta-\zeta_0)^2}{2\lambda\sigma^2}}e^{-\frac{\zeta^2}{2}}e^{-s^2+2s\zeta}d\zeta[/tex]

The last part is the generating function. Now in some simpler exercises I would get some part with [tex]e^s[/tex] and constants and Gauss integral, and then I would expand that part with Exp

[tex]\int_{-\infty}^\infty e^{-\frac{(\zeta-\zeta_0)^2}{2\lambda\sigma^2}}e^{-\frac{\zeta^2}{2}}e^{-s^2+2s\zeta}d\zeta=\sqrt{2\pi}\sqrt{\frac{\lambda\sigma^2}{\lambda\sigma^2+1}}e^{s^2-\frac{(\zeta_0-2s)^2}{2+2\lambda\sigma^2}}[/tex]

I have quadratic term in s and I would have different expansion for those parts with s and s^2, and I couldn't compare that to get the desired result :\

Can anyone point me in the right direction?

Last edited by a moderator: