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## Homework Statement

A particle moving in one dimensions is in the state [itex]|\psi\rangle[/itex] with position-space wave function [itex]\psi(x) = Ae^{−\lambda|x|}[/itex] where A, λ are positive real constants.

a)Normalize the wavefunction.

b)Determine the expectation values of x and [itex]x^2[/itex]

## Homework Equations

[tex]\langle\psi | \psi\rangle=1\][/tex]

[tex]\langle \hat{A}\rangle = \langle \psi |\hat{A}|\psi \rangle[/tex]

## The Attempt at a Solution

I used the first equation to normalize the wave function by doing

[tex]\int^{\infty}_{-\infty}A^2e^{-2\lambda |x|}dx[/tex]. I had to do this by splitting the integral into two parts to get rid of the absolute value. I ended up with [itex]A=\sqrt{\lambda}[/itex]

Then I got [itex]\langle x \rangle[/itex] by doing

[tex]\int^{\infty}_{-\infty}\lambda x e^{-2\lambda |x|}dx[/tex]which I had to use an integration by parts (one question I have is if there is an "easy" way to do IBP without listing out all of the variable changes and stuff. it's very time consuming. Anyways the answer I got is 0.

For [itex]x^2[/itex]I am trying to just plug it into the formula. The problem is I cannot seem to integrate this properly. I can plug it into mathematica but I cannot seem to work out the integral!