Follow these steps to show that solutions to Kummer's Equation (3.7.46) can be written in terms of Laguerre polynomials Ln (x ), which are defined according to a generating function as

where 0

(a) Prove that L_n(0) = n ! and L₀(x) = 1.

(b) Differentiate g(x , t) with respect to x, show that

and find the first few Laguerre polynomials.

(c) Differentiate g(x , t) with respect to t and show that

(d) Now show that Kummer's Equation is solved by deriving

and associate n with the principal quantum number for the hydrogen atom.

Transcribed Image Text:

g(x,t) = e-xt/(1-t) 1-1 = Στ()*, n=0 n!