Help solving the TISE for the hydrogen atom

[loba333]

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

 

[TSny]

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?)

 

[loba333]

Of course, sorry that was a silly mistake. Also the potential should be negative. another clumsy mistake. Thanks TSny
I will recalculate now

 

[loba333]

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 I am 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

 

[loba333]

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 ?

 

[TSny]

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 $r^2 \; sin\theta \; dr \; d\theta \; d\phi \;$]

 

[loba333]

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 don't understand what I am 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

 

[TSny]

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