# Expectation value of aharmonic oscillator

1. Sep 12, 2008

### Orson1981

1. The problem statement, all variables and given/known data
I need to find the expectation value of x of an aharmonic oscillator of a given potential:

$$V_{(x)} = c x^2 - g x^3 - f x^4$$

2. Relevant equations
Two relevant equations:
First:
I'm using the partition function to find the expectation value

$$<x>= \frac { \int x Z dx}{ \int Z dx}$$ integrating from -inf to +inf

where

$$Z = e^{\frac {-V} {k_{b} T}}$$

Second:
The professor gave us the solution of

$$(\frac {3 g} {4 c^2}) k_{b} T$$

3. The attempt at a solution

So it comes down to solving this integral, which frankly, I don't have a clue how to do. I've tried factoring out x^2 and factoring the remain potential. But that didn't help with the integral. I've played with substitution, setting $$u = e^{-c*x^2}$$, but that doesn't seem to get me anywhere. I've also looked at integration by parts, but that just doesn't seem to fit.

I've also tried mathematica and google, neither of which helped, and both of which feel a bit like cheating.

Thanks.

note: Please forgive and feel free to correct any incorrect terms I use, I'm notoriously sloppy when it comes to proper definitions and I'm trying to tighten it up a bit.

Last edited: Sep 12, 2008