"Exercise 1. Verify wave function." 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) Hhat(f) = phat(phat(f)) / (2 m) + V f phat(f) = -i hbar d(f,x) V = m omega^2 x^2 / 2 check(i hbar d(psi(0),t) == Hhat(psi(0))) check(i hbar d(psi(1),t) == Hhat(psi(1))) check(i hbar d(psi(2),t) == Hhat(psi(2))) check(i hbar d(psi(3),t) == Hhat(psi(3))) check(i hbar d(psi(4),t) == Hhat(psi(4))) "ok"
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