Elementary Wave Functions

1. Oct 16, 2013

richyw

1. The problem statement, all variables and given/known data

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 $x^2$

2. Relevant equations

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

3. 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!

2. Oct 16, 2013

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.

3. Oct 16, 2013

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.