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Homework Help: Elementary Wave Functions

  1. Oct 16, 2013 #1
    1. The problem statement, all variables and given/known data

    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]

    2. Relevant equations

    [tex]\langle\psi | \psi\rangle=1\][/tex]
    [tex]\langle \hat{A}\rangle = \langle \psi |\hat{A}|\psi \rangle[/tex]

    3. 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!
  2. jcsd
  3. Oct 16, 2013 #2


    User Avatar
    Science Advisor
    Homework Helper

    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.
  4. Oct 16, 2013 #3
    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.
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