"Perturbation example" ca0 = a cb0 = b dca = -i/hbar Hab(t) exp(-i omega t) cb dcb = -i/hbar Hba(t) exp(i omega t) ca "Verify equation (1)" dca1 = eval(dca, cb, cb0) dcb1 = eval(dcb, ca, ca0) check(dca1 == -i b / hbar Hab(t) exp(-i omega t)) check(dcb1 == -i a / hbar Hba(t) exp(i omega t)) "ok" "Verify equation (2)" ca1 = ca0 + integral(dca1,t) cb1 = cb0 + integral(dcb1,t) check(ca1 == a - i b / hbar integral(Hab(t) exp(-i omega t), t)) check(cb1 == b - i a / hbar integral(Hba(t) exp(i omega t), t)) "ok" "Verify equation (3)" dca2 = eval(dca, cb, cb1) dcb2 = eval(dcb, ca, ca1) Ta = -i b / hbar Hab(t) exp(-i omega t) - a / hbar^2 Hab(t) exp(-i omega t) * integral(Hba(t) exp(i omega t), t) Tb = -i a / hbar Hba(t) exp(i omega t) - b / hbar^2 Hba(t) exp(i omega t) * integral(Hab(t) exp(-i omega t), t) check(dca2 == Ta) check(dcb2 == Tb) "ok" "Verify equation (4)" ca2 = ca0 + integral(dca2,t) I = integral(Hba(t) exp(i omega t), t) T = a - i b / hbar integral(Hab(t) exp(-i omega t), t) - a / hbar^2 integral(Hab(t) exp(-i omega t) I, t) check(ca2 == T) "ok" "Verify equation (5)" cb2 = cb0 + integral(dcb2,t) I = integral(Hab(t) exp(-i omega t), t) T = b - i a / hbar integral(Hba(t) exp(i omega t), t) - b / hbar^2 integral(Hba(t) exp(i omega t) I, t) check(cb2 == T) "ok" Run