"Tunneling" psi1 = A exp(i k x) + B exp(-i k x) psi2 = C exp(i beta x) + D exp(-i beta x) psi3 = F exp(i k x) "Verify C and D" C = (beta + k) F exp(i k a - i beta a) / (2 beta) D = (beta - k) F exp(i k a + i beta a) / (2 beta) check(eval(psi2,x,a) == eval(psi3,x,a)) check(eval(d(psi2,x),x,a) == eval(d(psi3,x),x,a)) "ok" "Verify A and B" A = (k + beta) C exp( i k a - i beta a) / (2 k) + (k - beta) D exp( i k a + i beta a) / (2 k) B = (k - beta) C exp(-i k a - i beta a) / (2 k) + (k + beta) D exp(-i k a + i beta 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)) "ok" "Verify equation (5)" check(A / F == (k + beta)^2 exp(2 i a (k - beta)) / (4 k beta) - (k - beta)^2 exp(2 i a (k + beta)) / (4 k beta)) "ok" k = sqrt(2 m E) / hbar beta = sqrt(2 m (E - V0)) / hbar T1 = A/F conj(A/F) K = E / (V0 - E) + V0 / E + 1 "Verify equation (6)" check(T1 == 1 + K/8 (cos(4 a sqrt(2 m (E - V0)) / hbar) - 1)) "ok" "Verify equation (7)" check(T1 == 1 + K/4 sinh(2 i a / hbar sqrt(2 m (E - V0)))^2) "ok" "Verify equation (8)" check(T1 == 1 + V0^2 / (4 E (V0 - E)) sinh(2 i a / hbar sqrt(2 m (E - V0)))^2) "ok" "Verify equation (9)" joule = kilogram meter^2 / second^2 check(meter / (joule second) sqrt(kilogram joule) == 1) "ok" Run