"Atomic transitions 1" -- hydrogen wave function 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)) E(n) = -hbar^2 / (2 n^2 mu a0^2) -- for example, energy levels 1 and 2 na = 1 nb = 2 psia = psi(na,0,0) psib = psi(nb,0,0) Ea = E(na) Eb = E(nb) Psi = ca(t) psia exp(-i/hbar Ea t) + cb(t) psib exp(-i/hbar Eb t) -- time-independent Hamiltonian H0(f) = -hbar^2 D(f) / (2 mu) - hbar^2 / (mu a0 r) f -- Laplacian D(f) = 1/r^2 d(r^2 d(f,r), r) + 1/(r^2 sin(theta)) d(sin(theta) d(f,theta), theta) + 1/(r sin(theta))^2 d(f,phi,2) -- left side of Schrodinger equation A = i hbar d(Psi, t) -- right side of Schrodinger equation B = H0(Psi) + H1(Psi) -- C is the part that cancels C = Ea ca(t) psia exp(-i/hbar Ea t) + Eb cb(t) psib exp(-i/hbar Eb t) -- D is the part that remains D = i hbar d(ca(t),t) psia exp(-i/hbar Ea t) + i hbar d(cb(t),t) psib exp(-i/hbar Eb t) "Verify equation (1)" check(A - C == D) check(B - C == H1(Psi)) "ok" Run