# Hydrogen atom

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!

Related Advanced Physics Homework Help News on Phys.org
Meir Achuz
Homework Helper
Gold Member
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).

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!

Meir Achuz
Homework Helper
Gold Member
eku_girl83 said:
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?
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.

Meir Achuz
Homework Helper
Gold Member
eku_girl83 said:
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??
QUOTE]
The usual Legendre polynomials are normalized so that P_L(1)=1 for each value of L. No integral is involved. This is different (and easier) than the usual normalization of functions. For your problem, the normalization may not be necessary.

Meir Achuz