# Expected value of x for quantum oscillator - integration help

1. May 5, 2013

### phosgene

1. The problem statement, all variables and given/known data

I have a wavefunction $Cxe^{-ax^2}$ and I have to find the expected value of x.

2. Relevant equations

$∫_{-∞}^{∞} x^3 e^{-Ax^2} dx = 1/A^2 for A>0$

3. The attempt at a solution

I get an integral like this:

$<x>=|C|^2 ∫_{-∞}^{∞} x^3 e^{-Ax^2} dx$

After trying integration by parts and failing (miserably), I took the coward's way out and went to an integration table. I found this:

$∫_{-∞}^{∞} x^3 e^{-Ax^2} dx = 1/A^2 for A>0$

My question is this: my $a=ωm/ \hbar$ is always positive, right? I googled stuff on negative frequency, but I don't think that it applies in this situation (quantum oscillator).

Last edited: May 5, 2013
2. May 5, 2013

### phosgene

Actually, I think I got it using integration by parts, but I'd still like to know if my 'shortcut' is still valid.

3. May 6, 2013

### Office_Shredder

Staff Emeritus
If A<0, it's not that you need a different formula, it's that your integral doesn't converge at all.

Although for this specific integral I think you either wrote it down wrong or did the problem wrong because you're integrating an odd function and should just get 0 regardless of what a is (again, as long as it's positive so the integral converges)

4. May 6, 2013

### phosgene

I realized that now using Wolfram Alpha...that was my first thought, but then I thought that it would be too easy. I need to stop convincing myself that an easy solution is wrong solution...