"Hydrogen radius"

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

-- associated Laguerre polynomial (k is a local var)

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

-- spherical harmonic

Y(l,m) = (-1)^m sqrt((2 l + 1) / (4 pi) (l - m)! / (l + m)!) *
         P(l,m) exp(i m phi)

-- associated Legendre of cos theta (arxiv.org/abs/1805.12125)

P(l,m,k) = test(m < 0, (-1)^m (l + m)! / (l - m)! P(l,-m),
           (sin(theta)/2)^m sum(k, 0, l - m,
           (-1)^k (l + m + k)! / (l - m - k)! / (m + k)! / k! *
           ((1 - cos(theta)) / 2)^k))

"Verify equation (1)"

for(n, 1, 3, for(l, 0, n - 1, for(m, -l, l,
 a = psi(n,l,m),
 y = conj(a) a r^2 sin(theta),
 f = defint(expform(y), theta, 0, pi, phi, 0, 2 pi),
 I = integral(r f, r),
 rbar = 0 - eval(I,r,0),
 check(rbar == (3 n^2 - l (l + 1)) / 2 a0)
)))

"ok"