Elementary Wave Functions

  • Thread starter richyw
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  • #1
richyw
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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!
 

Answers and Replies

  • #2
dextercioby
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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
Bryson
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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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