psi1 = A exp(i k x) + B exp(-i k x) psi2 = C exp(i kappa x) + D exp(-i kappa x) psi3 = F exp(i k x) k = sqrt(2 m E) / hbar kappa = sqrt(2 m (E - V0)) / hbar "Verify C and D" C = (kappa + k) F exp(i k a - i kappa a) / (2 kappa) D = (kappa - k) F exp(i k a + i kappa a) / (2 kappa) check(eval(psi2,x,a) == eval(psi3,x,a)) check(eval(d(psi2,x),x,a) == eval(d(psi3,x),x,a)) check(-hbar^2 / (2 m) d(psi2,x,x) + V0 psi2 == E psi2) "ok" "Verify A and B" A = (k + kappa) C exp( i k a - i kappa a) / (2 k) + (k - kappa) D exp( i k a + i kappa a) / (2 k) B = (k - kappa) C exp(-i k a - i kappa a) / (2 k) + (k + kappa) D exp(-i k a + i kappa a) / (2 k) check(eval(psi1,x,-a) == eval(psi2,x,-a)) check(eval(d(psi1,x),x,-a) == eval(d(psi2,x),x,-a)) check(-hbar^2 / (2 m) d(psi1,x,x) == E psi1) "ok" T = A/F conj(A/F) K = E / (V0 - E) + V0 / E + 1 "Verify equation (5)" check(T == 1 + K/8 (cos(4 a sqrt(2 m (E - V0)) / hbar) - 1)) "ok" "Verify equation (6)" check(T == 1 + K/4 sinh(2 i a / hbar sqrt(2 m (E - V0)))^2) "ok" "Verify equation (7)" check(K == V0^2 / (E (V0 - E))) "ok" Run