1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Solutions of the Schrodinger equation for hydrogen

  1. Nov 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Consider an electron in the hydrogen atom with radial wave function [tex]R_{31}[/tex] (n=3, l=1). Please verify that this radial function verifies the radial equation.

    2. Relevant equations
    The radial equation

    [tex]\frac{1}{r^{2}}[/tex][tex]\frac{d}{dr}[/tex][tex]\left(r^{2}\frac{dR}{dr}\right)[/tex] + [tex]\frac{2\mu}{h^{2}}[/tex][tex]\left[E-V-\frac{h^{2}}{2\mu}\frac{l\left(l+1\right)}{r^{2}}\right][/tex]R = 0

    3. The attempt at a solution

    Ok so I found the corresponding solution for the given radial wave funtion, and I think I'm supposed to set that equal to A, some constant, times [tex]e^{\frac{-r}{3a_{0}}}[/tex]
    and then plug that into the original radial wave function? I'm not really sure of what I'm supposed to do here.
    Last edited: Nov 2, 2009
  2. jcsd
  3. Nov 2, 2009 #2
    Oh and those h's are supposed to be h bars. I don't know how to do that in latex.
  4. Nov 2, 2009 #3


    User Avatar
    Homework Helper
    Gold Member

    Just use the equation for [itex]R_{31}[/itex] that is in your text/notes, and substitute it into the Differential equation...

    P.S. To write [itex]\hbar[/itex] in [itex]\LaTeX[/itex], just use \hbar
  5. Nov 2, 2009 #4
    You mean like an equation like this?


    I tried using that and plugging it into the radial equation, but it gets really messy and I'm not sure if I know how to simplify it. I also don't know what to do with the V and E quantities.
  6. Nov 2, 2009 #5
    And I assumed that the stuff not depending on R was equal to some constant A to help make it easier... would that screw my answer up?
  7. Nov 2, 2009 #6


    User Avatar
    Homework Helper
    Gold Member


    [itex]V[/itex] is just the Coulomb potential, and if the electron is in the [itex]n=3[/itex] state, shouldn't [itex]E[/itex] be [itex]E_3[/itex] (which you should have an equation for)?
  8. Nov 2, 2009 #7
    Ok so [tex]V =\frac{1}{4\pi\epsilon_{0}}\frac{-e^{2}}{r}[/tex] and E is just [tex]\frac{-E_{0}}{n^{2}}[/tex]? And that will all cancel out if I plug everything in?
  9. Nov 2, 2009 #8


    User Avatar
    Homework Helper
    Gold Member

  10. Nov 2, 2009 #9
    Wow. Ok. Thank you so much! You've saved me a great deal of work.
  11. Nov 2, 2009 #10
    Also, try finding [tex] \frac {2\mu V}{\hbar ^2} [/tex] and [tex] \frac {2\mu E_n}{\hbar^2}[/tex] in terms of [itex]a_0[/itex] and [itex]r[/itex]. It might make it easier.
  12. Nov 2, 2009 #11
    Just to verify that this is correct, I'm getting [tex]\frac{2\mu V}{\hbar^{2}}=\frac{-2}{a_{0}r}[/tex] and [tex]\frac{2\mu E}{\hbar^{2}}=\frac{-1}{9a_{0}^{2}}[/tex] I'm getting almost everything to cancel out, but not quite everything. There might be an error in my derivatives.
  13. Nov 2, 2009 #12
    Ok I just got the answer. Thank you all so much for your help!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook