How to Calculate the Expectation Value of x^2 in Quantum Mechanics?

[jakeim86]

Homework Statement

I am given ψ(x), want to calculate <x^{2}>.

Homework Equations

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

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.

 

[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.

 

[squigglywolf]

then look up a table or formula for integrals of gaussians

 

[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?

 

[jakeim86]

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