"Time-independent Schrodinger equation" -- harmonic oscillator H(f) = -hbar^2 / (2 m) d(f,x,x) + 1/2 m omega^2 x^2 f psi(n) = C(n) 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) psi0 = psi(0) psi1 = psi(1) psi2 = psi(2) psi3 = psi(3) psi4 = psi(4) E(n) = hbar omega (n + 1/2) E0 = E(0) E1 = E(1) E2 = E(2) E3 = E(3) E4 = E(4) xi(n) = exp(-i E(n) t / hbar) xi0 = xi(0) xi1 = xi(1) xi2 = xi(2) xi3 = xi(3) xi4 = xi(4) Psi = (xi2 psi2 + xi3 psi3) / sqrt(2) check(i hbar d(Psi,t) == H(Psi)) -- integral of |f|^2 from minus infinity to plus infinity I(f) = 2 eval(integral(conj(f) f, x), exp(-m omega x^2 / hbar), 0, -- subst 0 for this exp erf(sqrt(m omega / hbar) x), 1) -- subst 1 for this erf check(I(Psi) == 1) -- time independent Schrodinger equation check((E2 psi2 + E3 psi3) / sqrt(2) == H((psi2 + psi3) / sqrt(2))) "ok" Run