"Harmonic oscillator 2" psi(n) = (-1)^n C(n) exp(m omega x^2 / (2 hbar)) * 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) psi0 = psi(0) psi1 = psi(1) psi2 = psi(2) psi3 = psi(3) psi4 = psi(4) "Verify step-up operator" ahat1(f) = sqrt(hbar / (2 m omega)) (f m omega x / hbar - d(f,x)) check(ahat1(psi0) == sqrt(1) psi1) check(ahat1(psi1) == sqrt(2) psi2) check(ahat1(psi2) == sqrt(3) psi3) check(ahat1(psi3) == sqrt(4) psi4) "ok" "Verify step-down operator" ahat(f) = sqrt(hbar / (2 m omega)) (f m omega x / hbar + d(f,x)) check(ahat(psi1) == sqrt(1) psi0) check(ahat(psi2) == sqrt(2) psi1) check(ahat(psi3) == sqrt(3) psi2) check(ahat(psi4) == sqrt(4) psi3) "ok" "Verify number operator" N(f) = ahat1(ahat(f)) check(N(psi0) == 0 psi0) check(N(psi1) == 1 psi1) check(N(psi2) == 2 psi2) check(N(psi3) == 3 psi3) check(N(psi4) == 4 psi4) "ok" Run