Hydrogen Radial Equation: Recursion Relation & Laguerre Polynomials

[jayqubee]

I'm in the first of 3 courses in quantum mechanics, and we just started chapter 4 of Griffiths. He goes into great detail in most of the solution of the radial equation, except for one part: translating the recursion relation into a form that matches the definition of the Laguerre polynomials. Now I understand the technique that gets all the way to that point, but I have yet to find any derivation that actually shows how that recursion relation is made to match the right form. I spent a while on it but I can't get it quite right. Can anyone show, or point me to a derivation that includes this detail? Thanks.

 

[dextercioby]

And what definition of Laguerre's polynomials do you use ?

 

[jayqubee]

The recursion relation i have is:

c_j+1= \frac{j+L+1-n}{(j+1)(j+2L+2)}c_j

for principle quantum number n, and orbital quantum number L, where the coefficients terminate after
j_max = n - L - 1

The definition I'm trying to match is the one in Arfken & Weber:

L^{k}_{N} = \sum^{N}_{j=0}\frac{(-1)^j(N + k)!}{j!(N-j)!(k+j)!}x^j

N = j_max = n - L - 1

k = 2L + 1

 

[Bill_K]

A number of documents on the web follow Griffiths' treatment, and try to add some explanation, especially at this point. The best one I've found is here.

The problem, as dextercioby says, lies in how you define the Laguerre polynomials. There are three ways you can do it: the DE, the recursion relation, and Rodrigues' formula (which is what Griffiths uses in Eqs. 4.87 and 4.88). Any one of these can be tied to the radial wavefunction for the hydrogen atom. The third is the least useful. and very difficult to derive from the others. The ref I gave above defines them via the DE and derives the recursion relation from that.

 

[dextercioby]

I find this treatment better: http://www.physics.drexel.edu/~tim/open/hydrofin/node6.html

 

[jayqubee]

Wow it is so much easier to manipulate the \upsilon(\rho) equation into the associated Laguerre ODE than to use the recursion relation. Thanks for your help!