This problem is killing me.

$\displaystyle \Psi(x) = Ae^{-\frac{(x-x_0)^2}{2K^2}}$

I normalized the wave packet

$\displaystyle \int^\infty_{-\infty} \Psi\Psi*\,dx = 1 $ where * is the complex conjugate

using a table integral:

$\displaystyle \int^{\infty}_{-\infty}{e^{-\frac{(x-b)^2}{c^2}}}\,dx=c\sqrt{\pi}$

$\displaystyle A^2 = \frac{1}{K\sqrt{\pi}}$

Now I need the expected value of x

$\displaystyle <x>=\int^\infty_{-\infty} x\Psi\Psi*\,dx$

$\displaystyle =A^2\int^\infty_{-\infty} xe^{-\frac{(x-x_0)^2}{K^2}}\,dx$

which I expect to be $\displaystyle x_0$

but I can't get it to work out. I tried substitution with u = the exponent but looking back over it I don't think it will work right because I can't seem to get rid of all the references to x when I substitute.

Any pointers?