"Fermi's golden rule"

"Verify equation (1)"

Psi = c1(t) psi1(r) exp(-i omega1 t) +
      c2(t) psi2(r) exp(-i omega2 t)

check(i hbar d(Psi, t) ==
i hbar d(c1(t),t) psi1(r) exp(-i omega1 t) +
hbar omega1 c1(t) psi1(r) exp(-i omega1 t) +
i hbar d(c2(t),t) psi2(r) exp(-i omega2 t) +
hbar omega2 c2(t) psi2(r) exp(-i omega2 t))

"ok"

"Verify equation (2)"

Delta = omega2 - omega1

f = cos(omega t + phi) exp(i Delta t)
I = defint(expform(f), t, 0, t)

c2 = 2 M21 / (i hbar) I

T1 = -M21 / hbar *
     (exp(i (Delta - omega) t) - 1) / (Delta - omega) *
     exp(-i phi)

T2 = -M21 / hbar *
     (exp(i (Delta + omega) t) - 1) / (Delta + omega) *
     exp(i phi)

check(c2 == T1 + T2)

"ok"

"Verify equation (3)"
check(exp(i x) - 1 == 2 i sin(x/2) exp(i x/2))
"ok"

"Verify equation (4)"

c2 = -M21 / hbar *
     (exp(i (Delta - omega) t) - 1) / (Delta - omega) *
     exp(-i phi)

sinc(x) = sin(x) / x

check(c2 == -i t M21 / hbar sinc(1/2 (Delta - omega) t) exp(1/2 i (Delta - omega) t) exp(-i phi))

"ok"

"Verify equation (5)"

P = conj(c2) c2

check(P == t^2 / hbar^2 M21^2 sinc((Delta - omega) / 2 t)^2)

"ok"