# Homework Help: Expectation value of x^2 (Q.M.)

1. Jan 27, 2012

### jakeim86

1. The problem statement, all variables and given/known data
I am given ψ(x), want to calculate $<x^{2}>$.

2. Relevant equations
$\psi(x) = a\exp(ibx-(c/2)(x-d)^2)$
$<x^2> = \int\limits_{-∞}^∞ \psi^*x^2\psi \mathrm{d}x$

3. The attempt at a solution
Well, I normalized the wave function and found $a = (\frac{c}{\pi})^{1/4}$.
So, the integral I have to do becomes:
$<x^2> = \sqrt{\frac{c}{\pi}} \int\limits_{-∞}^∞ x^2\exp{(-c(x-d)^2)}\mathrm{d}x$.

Since the function is neither even nor odd, there is no simple trick, and I had a hard time finding the exact integral in my table of integrals.

Thanks in advance.

2. Jan 27, 2012

### Clever-Name

I would suggest a change of variables. Try u = x-d

That should put it in a form where you could split it up then use an integral table.

3. Jan 28, 2012

### squigglywolf

then look up a table or formula for integrals of gaussians

4. Jan 28, 2012

### ShadowDatsas

Yeap, after you use y = x -d and split the integral in three: y^2*exp(ay) and d^2*exp(ay) and 2yd*exp(ay) its very easy. The first one gives (π/α)^(3/2) and the second one (π/α)^(1/2). The third one can be found through "derivative integration" if I call it correclty in english?

5. Jan 28, 2012

### jakeim86

Ah, a simple u-substitution works. Thanks for the help everyone.

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