"Harmonic oscillator coherent state" psi(n) = K(n) exp(m omega / (2 hbar) (x - xbar)^2) * exp(i / hbar pbar (x - xbar / 2)) * exp(-i (n + 1/2) omega t) * d(exp(-m omega (x - xbar)^2 / hbar), x, n) K(n) = (-1)^n / sqrt(2^n n!) * (m omega / (pi hbar))^(1/4) * (hbar / (m omega))^(n/2) xbar = sqrt(2 hbar / m / omega) r cos(omega t + theta) pbar = -sqrt(2 m hbar omega) r sin(omega t + theta) psi0 = psi(0) psi1 = psi(1) psi2 = psi(2) psi3 = psi(3) psi4 = psi(4) Hhat(f) = phat(phat(f)) / (2 m) + V f phat(f) = -i hbar d(f,x) V = m omega^2 x^2 / 2 "Verify wave functions" check(i hbar d(psi0,t) == Hhat(psi0)) check(i hbar d(psi1,t) == Hhat(psi1)) check(i hbar d(psi2,t) == Hhat(psi2)) check(i hbar d(psi3,t) == Hhat(psi3)) check(i hbar d(psi4,t) == Hhat(psi4)) "ok" A = m omega / hbar B = 2 sqrt(2 m omega / hbar) r cos(omega t + theta) C = -2 r^2 cos(omega t + theta)^2 G2 = sqrt(pi / A) / (2 A) (1 + B^2 / (2 A)) exp(B^2 / (4 A) + C) G1 = sqrt(pi / A) (B / (2 A)) exp(B^2 / (4 A) + C) G0 = sqrt(pi / A) exp(B^2 / (4 A) + C) -- X is expectation of x f = conj(psi0) x psi0 C1 = sqrt(m omega / (hbar pi)) check(f == C1 x exp(-A x^2 + B x + C)) X = C1 G1 -- gaussian integral -- X2 is expectation of x^2 f = conj(psi0) x^2 psi0 C2 = sqrt(m omega / (hbar pi)) check(f == C2 x^2 exp(-A x^2 + B x + C)) X2 = C2 G2 -- gaussian integral -- P is expectation of p f = conj(psi0) phat(psi0) C1 = i (m omega)^(3/2) / sqrt(pi hbar) C0 = -sqrt(2 / pi) m omega r * (sin(omega t + theta) + i cos(omega t + theta)) check(f == (C1 x + C0) exp(-A x^2 + B x + C)) P = C1 G1 + C0 G0 -- gaussian integral -- P2 is expectation of p^2 f = conj(psi0) phat(phat(psi0)) C2 = -sqrt(m^5 omega^5 / (pi hbar)) C1 = 2 sqrt(2 / pi) m^2 omega^2 r cos(omega t + theta) - 2 sqrt(2 / pi) i m^2 omega^2 r sin(omega t + theta) C0 = sqrt(hbar m^3 omega^3 / pi) * (1 - 2 r^2 cos(omega t + theta)^2 + 2 r^2 sin(omega t + theta)^2 + 4 i r^2 cos(omega t + theta) sin(omega t + theta)) check(f == (C2 x^2 + C1 x + C0) exp(-A x^2 + B x + C)) P2 = C2 G2 + C1 G1 + C0 G0 -- gaussian integral "Verify uncertainty" check(X2 - X^2 == hbar / (2 m omega)) check(P2 - P^2 == m hbar omega / 2) check(sqrt((X2 - X^2) (P2 - P^2)) == hbar / 2) "ok" Run