"Exercise 2. Verify normalization." psi(n) = 1 / sqrt(2^n n!) * (m omega / (pi hbar))^(1/4) * H(n, sqrt(m omega / hbar) (x - xbar)) * exp(-m omega / (2 hbar) (x - xbar)^2) * exp(i / hbar pbar (x - xbar / 2)) * exp(-i (n + 1/2) omega t) H(n,y,z) = (-1)^n exp(y^2) eval(d(exp(-z^2),z,n),z,y) xbar = sqrt(2 hbar / m / omega) r cos(omega t + theta) pbar = -sqrt(2 m hbar omega) r sin(omega t + theta) f = conj(psi(1)) psi(1) 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) C2 = 2 m^(3/2) omega^(3/2) hbar^(-3/2) pi^(-1/2) C1 = (-4) sqrt(2 / pi) m omega r / hbar cos(omega t + theta) C0 = 4 sqrt(m omega / (pi hbar)) r^2 cos(omega t + theta)^2 check(f == (C2 x^2 + C1 x + C0) exp(-A x^2 + B x + C)) I = C2 G2 + C1 G1 + C0 G0 -- gaussian integral check(I == 1) "ok"
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