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Normalizing the wave function of the electron in hydrogen

  1. Apr 1, 2017 #1
    1. The problem statement, all variables and given/known data
    upload_2017-4-1_19-3-55.png
    I am having trouble with part d, where they ask me to prove that the wave function is already normalized

    3. The attempt at a solution
    upload_2017-4-1_19-5-18.png
    But that clearly doesn't give me 1. I tried to use spherical coordinates since it is in 3D? Not really sure how to proceed.
    EDIT: I realize that I didn't square the wave function, so
    upload_2017-4-1_19-8-5.png
    Which still doesn't give me 1
     
  2. jcsd
  3. Apr 1, 2017 #2

    blue_leaf77

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    The integration limit should be from ##0## to ##\infty##.
     
  4. Apr 1, 2017 #3
    Ok. I was thinking 0 to r since the most it could go was to the radius. I guess the logic is wrong
    upload_2017-4-1_19-47-25.png
    multiplied by 4pi, so the 4's will cancel out, leaving me with (a_0^3)(pi)
    Which still doesn't = 1. What am I missing
     
  5. Apr 1, 2017 #4

    kuruman

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    It pays to be methodical
    ##\psi(r,\theta,\phi) =\frac{1}{\sqrt{\pi}} \left( \frac{1}{a_0} \right )^{3/2} e^{-r/a_0}##
    ##\psi^*(r,\theta,\phi) \psi(r,\theta,\phi) =\frac{1}{{\pi}} \left( \frac{1}{a_0} \right )^{3} e^{-2r/a_0}##
    Now do the integrals.
     
  6. Apr 2, 2017 #5
    If I use the spherical coordinates I still get (a^3)π. I say that 0<Θ<π and 0<Φ<2π
     
  7. Apr 2, 2017 #6

    kuruman

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    Can you show the expression that you integrated to get this result?
     
  8. Apr 2, 2017 #7
    upload_2017-4-2_19-0-21.png
    I took out the constant and then used spherical coordinates
     
  9. Apr 2, 2017 #8
    Oh wait. I think I figured it out. I lost my constant along the way.
    When I integrate to find the probability of finding the electron in a certain place will I use spherical coordinates again ?
     
  10. Apr 2, 2017 #9

    kuruman

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    Yes.
     
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