"Radial eigenfunctions"

clear

e = quote(e)

R(n,l) = 2 / (n^2 a0^(3/2)) *
         sqrt((n - l - 1)! / (n + l)!) *
         (2 rho / n)^l *
         L(2 rho / n,n - l - 1,2 l + 1) *
         exp(-rho / n)

L(x,m,j) = sum(k,0,m,(-x)^k (m + j)! / ((m - k)! (j + k)! k!))

rho = r / a0
A = a0^(3/2)

check(R(1,0) A == 2 exp(-rho))
check(R(2,0) A == sqrt(2)/4 (2 - rho) exp(-rho/2))
check(R(2,1) A == sqrt(6)/12 rho exp(-rho/2))
check(R(3,0) A == 2 sqrt(3)/27 (3 - 2 rho + 2/9 rho^2) exp(-rho/3))
check(R(3,1) A == sqrt(6)/81 rho (4 - 2/3 rho) exp(-rho/3))
check(R(3,2) A == 2 sqrt(30)/1215 rho^2 exp(-rho/3))
