"Ehrenfest theorem" -- harmonic oscillator psi(n) = C(n) exp(-i E(n) t / hbar) * exp(m omega x^2 / (2 hbar)) * d(exp(-m omega x^2 / hbar), x, n) C(n) = (-1)^n / sqrt(2^n n!) * (m omega / (pi hbar))^(1/4) * (hbar / (m omega))^(n/2) E(n) = hbar omega (n + 1/2) -- Psi has nonzero expectation for p and x Psi = (psi(1) + psi(2)) / sqrt(2) EXP = exp(-m omega x^2 / hbar) ERF = erf(sqrt(m omega / hbar) x) -- check normalization of Psi I = integral(conj(Psi) Psi, x) -- evaluate I from minus infinity to plus infinity I = eval(I, EXP, 0, ERF, 1) - eval(I, EXP, 0, ERF, -1) check(I == 1) -- xbar is expectation value for x I = integral(conj(Psi) x Psi, x) -- evaluate I from minus infinity to plus infinity xbar = eval(I, EXP, 0, ERF, 1) - eval(I, EXP, 0, ERF, -1) xbar -- pbar is expectation value for p I = integral(conj(Psi) (-i hbar) d(Psi,x), x) -- evaluate I from minus infinity to plus infinity pbar = eval(I, EXP, 0, ERF, 1) - eval(I, EXP, 0, ERF, -1) pbar "Verify momentum formula" check(pbar == m d(xbar,t)) "ok" Run