Vibrational Motion - Calculating Mean Square Displacement

[Shiz]

Homework Statement

Calculate the mean square displacement x² of the particle from its
equilibrium position.

Homework Equations

∫ from -\infty to +\infty of N_v² * H_v(y) * e^{-y^2} dy

Since y=x/\alpha, \alphady=dx

yH_v = vH_v-1 + (1/2)H_v+1

The Attempt at a Solution

https://www.dropbox.com/s/uiqbgzjjlqnnqwk/2014-02-14%2022.23.41.jpg

What is boxed is where I distributed everything. That looked horrible so I applied the recursion relation once. I believe the last integral goes to zero. Integrating an odd function over a symmetrical range would be zero. But then everything would be zero and that's just wrong. There should be a relationship between v and m and k_f and hbar from \alpha. I apologize, but please explain the math as simple as possible. The math is the issue, not really the concept.

EDIT: Should I apply the recursion relation once more in the integral as it has yH_v?

 

[ShayanJ]

I don't know that much about Hermite polynomials to judge your calculations but what I know is that they are alternating in being odd and even.In fact even numbered ones are even and odd numbered ones are odd.
Take this into account when calculating the integral!

 

[ShayanJ]

Equation (50) in this page may help too.