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Homework Help: Radial Schrödinger Equation

  1. Dec 11, 2009 #1
    Sorry in advance my english, I tried to translate it to english as good as I can

    1. The problem statement, all variables and given/known data

    When [tex]l[/tex] has its maximum value, the hydrogen atom radial equation has a simple form of

    R(r) = Arn-1e-r/na0, where a0 is Bohr's radius.

    Write the respective radial schrödinger equation.



    2. Relevant equations

    Radial Schrödinger Equation:
    [tex]-\frac{\hbar^2}{2*\mu*r^2}*\frac{d}{dr}(r^2*\frac{dR(r)}{dr})+\left[-\frac{kZe^2}{r}+\frac{\hbar^2*l(l+1)}{2*\mu*r^2}\right]*R(r) = ER(r)[/tex]


    3. The attempt at a solution

    I've attempted substituting the given R(r) to the radial equation, but I cant get anything out of it that makes sense. Too long to write it here with this hard latex thing :S

    Should I get some nice reduced form or can I expect some equation monster?
     
  2. jcsd
  3. Dec 12, 2009 #2

    gabbagabbahey

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    Your translation is very good!:smile:

    This is the key assumption in the problem statement....What is the maximum value of [itex]l[/itex] for any given value of [itex]n[/itex] (remember, only certain values are allowed)?....Substitute that into your radial Schroedinger equation (along with the given function [itex]R(r)[/itex]).


     
  4. Dec 12, 2009 #3
    Thanks for answer!
    The maximum value of [itex]l[/itex] is [itex]n-1[/itex] right?
    I tried to substitute it but it doesnt get much prettier :)

    I'm doing the math with Maple here, and it freaks me out to even think that I should be able to do it without computer, which is the case actually...That's why I think I'm doing something wrong, maybe I understand the equation wrong or something.

    Here's a screenshot of my maple session (left side of the equation)
    http://img526.imageshack.us/img526/5863/tryw.jpg [Broken]

    If I count the right side in too, I can reduce the Arn-1e-r/na0 term from both sides but still I'm not so sure about it
     
    Last edited by a moderator: May 4, 2017
  5. Dec 12, 2009 #4

    gabbagabbahey

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    You also have an equation for Bohr's radius [itex]a_0[/itex] right?...And [itex]Z=[/itex]____ for the Hydrogen atom?

    When you substitute these things in, you should get a lot of terms canceling each-other out.
     
  6. Dec 12, 2009 #5
    aaah the bohr radius equation, didnt even think about it :)
    (and Z=1 ofcourse)

    the next part of the question was to prove that the given R(r) is the solution to the schrödinger equation with [itex]l=0[/itex], and I got it right now with that bohr radius tip you gave! :)
    [tex]- \frac{1}{2} \frac{k^{2}e^{4}\mu}{\hbar ^{2}} = E[/tex]

    But with [itex]l=n-1[/itex] I still get horrible monster equations...*cry* nothing seems to cancel out

    there must be something I'm doing wrong since I cant be expected to do that kind of differentations without computer in a 2 hour exam !! No human could do it right! :D
    (this is an old exam question)
     
    Last edited: Dec 12, 2009
  7. Dec 12, 2009 #6
    Aaah I got it!
    0 = 0 finally :)
    There was some silly mistake in the equation.

    The derivation is still inhumane to do with pen and paper though :)

    Thanks much gabbagabbahey!
     
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