Help solving the TISE for the hydrogen atom

  • Thread starter loba333
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  • #1
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Homework Statement


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


Homework Equations



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


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
 

Answers and Replies

  • #2
TSny
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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?)
 
  • #3
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Of course, sorry that was a silly mistake. Also the potential should be negative. another clumsy mistake. Thanks TSny
I will recalculate now
 
  • #4
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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
 
  • #5
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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 ?
 
  • #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:
  • #7
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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
 
  • #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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