"Harmonic oscillator 1" psi(n) = C(n) exp(m omega x^2 / (2 hbar) - i omega t (n + 1/2)) * d(exp(-m omega x^2 / hbar), x, n) C(n) = 1 / sqrt(2^n n!) * (m omega / (pi hbar))^(1/4) * (hbar / (m omega))^(n/2) "Verify wave functions solve Schrodinger equation" psi0 = psi(0) psi1 = psi(1) psi2 = psi(2) psi3 = psi(3) psi4 = psi(4) Hhat(f) = -hbar^2 / (2 m) d(f,x,x) + 1/2 m omega^2 x^2 f 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" "Verify for time-independent Schrodinger equation" psi(n) = C(n) exp(m omega x^2 / (2 hbar)) * d(exp(-m omega x^2 / hbar), x, n) psi0 = psi(0) psi1 = psi(1) psi2 = psi(2) psi3 = psi(3) psi4 = psi(4) E(n) = hbar omega (n + 1/2) check(Hhat(psi0) == E(0) psi0) check(Hhat(psi1) == E(1) psi1) check(Hhat(psi2) == E(2) psi2) check(Hhat(psi3) == E(3) psi3) check(Hhat(psi4) == E(4) psi4) "ok" Run