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Homework Help: Expected r^2 for the 2s wavefunction of hydrogen atom

  1. Sep 6, 2010 #1
    1. The problem statement, all variables and given/known data
    Calculate the expected value for r^2 for the 2s wavefunction of the hydrogen atom (only the radial part of the function is needed for l=0). If you choose to solve this problem graphically, plot or sketch the function you integrate.


    2. Relevant equations
    R(r)=1/sqrt(2a^3)*(1-r/2a)*e^(-r/2a) where a=bohr radius



    3. The attempt at a solution
    to calculate, i know you integrate (from 0 to inf) as follows: int((R(r))^2*r^2 dr),
    but i'm having trouble solving the integral. I'm not sure how I would solve it graphically either.

    Any help is appreciated. Thanks
     
  2. jcsd
  3. Sep 6, 2010 #2

    vela

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    If you're familiar with the gamma function, you can use

    [tex]\Gamma(n) = \int_0^\infty t^{n-1}e^{-t}\,dt = (n-1)![/tex]

    If you need to show that result, you can integrate by parts to prove it by induction. Or you can use this trick:

    [tex]\int_0^\infty t^ne^{-\alpha t}\,dt = \int_0^\infty \left(-\frac{d}{d\alpha}\right)^n e^{-\alpha t}\,dt = \left(-\frac{d}{d\alpha}\right)^n \int_0^\infty e^{-\alpha t}\,dt[/tex]

    Do the integral, differentiate, and then set [itex]\alpha=1[/itex].
     
    Last edited: Sep 6, 2010
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