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Hydrogen atom

  1. Apr 9, 2006 #1
    I solved the differential equation for theta portion of the hydrogen wave function using a power series solution. I got a sub n+2 = a sub n ((n(n+1)-C)/(n+2)(n+1)). I then truncated the power series at n = l to get
    C= l(l+1).

    I know need to use the recursion formula I found to find the l = 0, 1, 2, and 3 solutions to the differential equation. Do I simply plug l in for n? If so, I get for l = 0, a2 = -Ca0/2. Is this the SOLUTION to the D.E. for
    l = 0, or do I need to do something else?
    Similarly, for l = 1, I get a3= a1 (2-C)/6.

    Any help appreciated!
     
  2. jcsd
  3. Apr 10, 2006 #2

    Meir Achuz

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    Put in C=L(L+1) and do the arith.
    The last step is to require P_L(1)=1, which is the normalization conditon for Legendre polynomials (the name of the theta solutions).
     
  4. Apr 10, 2006 #3
    If I sub in C= L(L+1), then I get a2 = -l(l+1)a0/2. But this is the solution for l = 0...so if I let l = 0, then I get a2 =0. This is also true for a3, a4, a5, .... Is this ok?

    Also...when I normalize...Do I do the integral of a2^2 from -1 to 1 = 1 (since x = cos theta) and solve for a0??

    I'm just a bit confused...thanks!
     
  5. Apr 10, 2006 #4

    Meir Achuz

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    Yes. For any L, the solution is a poynomial of degree L.
    For L=0, P_0=1. For L_1. P_1=1, etc.
     
  6. Apr 10, 2006 #5

    Meir Achuz

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  7. Apr 10, 2006 #6

    Meir Achuz

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    It may be time to look in a Math Physics book under
    Legendre Polynomials in the index.
     
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