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Help solving the TISE for the hydrogen atom

  1. Dec 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi guys the question is
    "Write down the time independent Schrödinger equation for the hydrogen atom,
    and show that the wave function
    Ψ(r,θ ,φ ) = Ae(−r / aB)
    is a solution. (A is a normalization constant and aB is the Bohr radius.) What is
    the energy of the state with this wave function?"

    I have solved what i think is the right answer, but i'm not to sure what else i have to do to it or if i have the right answer.

    Any help would be appreciated


    2. Relevant equations

    Heres the lectures online notes
    http://www.mark-fox.staff.shef.ac.uk/PHY332/phy332_notes.pdf


    3. The attempt at a solution
    Here is my attempt at the solution
    http://imgur.com/hMZq6
    And the paper it was taken from
    http://imgur.com/MSG1i
     
  2. jcsd
  3. Dec 27, 2012 #2

    TSny

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    Gold Member

    The energy should be a constant (independent of r). Don't forget the contribution from the potential energy term. (Is the potential energy positive or negative?)
     
  4. Dec 27, 2012 #3
    Of course, sorry that was a silly mistake. Also the potential should be negative. another clumsy mistake. Thanks TSny
    I will recalculate now
     
  5. Dec 30, 2012 #4
    So i'm getting the right answer for the energy now, even though the correct answer appears to have set A to 1 or its just not included in the final solution ?

    Also have i normalized the WF correctly. For the next part we have to find the expectation value of r and im getting an odd answer and feel like its something to do with my normalization constant.

    The correct answer should be 1.5aB. Any help would be really appreciated http://i.imgur.com/qgOmA.jpg?1
     
  6. Dec 30, 2012 #5
    just realized i didn't square a when finding the expectation value of r .

    Still it comes 0.5aB, any idea where the 3 comes from ?
     
  7. Dec 30, 2012 #6

    TSny

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    The constant ##A## should cancel out when finding the energy. In the fourth line of your notes, there should not be any factors of ##A## (because ##A## is already included in ##\psi##).

    When you normalize the wavefunction or when you want to find the expectation value of ##r##, you need to integrate over all three dimensions of space rather than just integrate over ##r##. So, you'll also need to integrate over the spherical coordinates ##\theta## and ##\phi##. But that should be easy since the wavefunction doesn't depend on those two variables. [Recall that the volume element in spherical coordinates is ##r^2 \; sin\theta \; dr \; d\theta \; d\phi \;##]
     
    Last edited: Dec 30, 2012
  8. Dec 30, 2012 #7
    Oh yeah thanks. I sorted that first one out.

    That makes sense to integrate over all space, how ever now i'm really confused as the integral has a r^3 value in it. it says you can use the identities at the bottom (http://imgur.com/MSG1i) but i swear they don't hold as it doesn't include a scenario for the exponential's exponent having a coefficient, which in out case we clearly do (-2/aB)

    Also i dont understand what im going to do with the value 4∏ as a result of the angular part of the volume element.

    Thank you very much for your time btw TSny
     
  9. Dec 30, 2012 #8

    TSny

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    Try the substitution x = 2r/aB in your integrals. The 4[itex]\pi[/itex] will just be a numerical factor that you will need to include. It will contribute something to the normalization constant A.
     
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