"Photoelectric effect" "Verify equation (1)" A = (Ax(),Ay(),Az()) PA = -i hbar div(A psi()) AP = dot(A, -i hbar grad(psi())) check(PA + AP == 2 AP - i hbar psi() div(A)) "ok" "Verify equation (2)" kr = kx x + ky y + kz z E = E0 sin(kr - omega t) (e1,e2,e3) A = -E0 / omega cos(kr - omega t) (e1,e2,e3) check(E == -d(A,t)) "ok" "Verify equation (3)" f = exp(i k r u - r/a) I = defint(f, u, -1, 1) check(I = (exp(i k r - r/a) - exp(-i k r - r/a)) / (i k r)) "ok" "Verify equation (4)" f = (exp(i k r - r/a) - exp(-i k r - r/a)) r^2 I = integral(f,r) I = 0 - eval(I,r,0) check(I == -2 / (i k - 1/a)^3 + 2 / (-i k - 1/a)^3) "ok" "Verify equation (5)" check(I/k == i a^4 (16 / (a^2 k^2 + 1)^3 - 4 / (a^2 k^2 + 1)^2)) "ok" "Verify equation (6)" M = -pi / sqrt(pi a^3 l^3) i a^4 check(M conj(M) == pi a^5 / l^3) "ok" "Verify equation (7)" Eb = hbar^2 k^2 / (2 m) m = hbar / (a alpha c) check(sqrt(2 m^3 Eb) / hbar^3 == k / (a alpha hbar c)) "ok" "Verify equation (8)" R = 2 pi / hbar M conj(M) rho rho = (l / (2 pi))^3 k / (a alpha hbar c) check(R == k a^4 / (4 pi alpha hbar^2 c)) "ok" "Verify equation (9)" joule = kilogram meter^2 / second^2 newton = kilogram meter / second^2 check(1/meter meter^4 / ((joule second)^2 meter/second) newton^2 == 1/second) "ok" Run