How Do You Normalize and Calculate Expectation Values in Quantum Mechanics?

[richyw]

Homework Statement

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

a)Normalize the wavefunction.

b)Determine the expectation values of x and $x^2$

Homework Equations

$$\langle\psi | \psi\rangle=1\]$$
$$\langle \hat{A}\rangle = \langle \psi |\hat{A}|\psi \rangle$$

The Attempt at a Solution

I used the first equation to normalize the wave function by doing
$$\int^{\infty}_{-\infty}A^2e^{-2\lambda |x|}dx$$. I had to do this by splitting the integral into two parts to get rid of the absolute value. I ended up with $A=\sqrt{\lambda}$

Then I got $\langle x \rangle$ by doing

$$\int^{\infty}_{-\infty}\lambda x e^{-2\lambda |x|}dx$$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 $x^2$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!

 

[dextercioby]

Surely A = Realthingy times e(i alpha), alpha is an arbitrary realthingy.

The method you used to compute the integral in <x> must work for <x^2> as well. You need to do partial integration not once, but twice.

 

[Bryson]

Do you need to use Mathematica? Just use integral table or or solve it as mentioned above. In all my QM courses, we never used Maple, or Mathematica.