"Hydrogen atom"

-- hydrogen atom eigenfunctions

clear

e = quote(e)

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 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

a0 = 4 pi epsilon0 hbar^2 / (e^2 mu)

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

P(y,l,m) = eval(1 / (2^l l!) (1 - x^2)^(m/2) *
           d((x^2 - 1)^l,x,l + m),x,y)

H(psi) = -hbar^2 Lap(psi) / (2 mu) - e^2 / (4 pi epsilon0 r) psi

Lap(f) = 1/r^2 d(r^2 d(f,r),r) +
         1/(r^2 expsin(theta)) d(expsin(theta) d(f,theta),theta) +
         1/(r expsin(theta))^2 d(f,phi,2)

E(n) = -mu / (2 n^2) (e^2 / (4 pi epsilon0 hbar))^2

S(n,l,m) = H(psi(n,l,m)) == E(n) psi(n,l,m)

check(S(1,0,0) == 1)
check(S(2,0,0) == 1)
check(S(2,1,0) == 1)
check(S(2,1,1) == 1)
check(S(2,1,-1) == 1)
check(S(3,0,0) == 1)
check(S(3,1,0) == 1)
check(S(3,1,1) == 1)
check(S(3,1,-1) == 1)
check(S(3,2,0) == 1)
check(S(3,2,1) == 1)
check(S(3,2,2) == 1)
check(S(3,2,-1) == 1)
check(S(3,2,-2) == 1)
