Assume that the wave approaches the barrier from the negative x direction. The coefficient B cannot be set equal to zero because Be^-I squareroot (2um E/h^2)x represents reflection from the barrier. However. G can be set equal to zero because there is no wave incident on the barrier from the positive x direction. a. The wave functions and their derivatives must be continues at x = 0 and x =

a. Show that the coefficients must satisfy the following conditions: A + B = C + D Ce^-ka + De^+ka = Fe^+ika A - B = - ik/k (-C + D) -C^-ka + De^ka = ik/k Fe^+ika b. Because the transmission probability is given by |IF/A|^2, it is useful to manipulate these equations to get a relationship between F and A. By adding and subtracting the first pair of equations. A and B can be expressed in terms of C and D. The second pair of equations can be combined in the same way to give equations for D and C in terms of F. Show that D = ik e^+ika + ke^ika/+ika/2k e^+ka C = - I k e^+ika + k e^+ika/2k e^ka F, and A = (ik - k)C + (ik + k)D/2ik
c. Substitute these results for C and Din terms of F in to A = (ik - k)C + (ik + k)D/2ik to relate A and F. Show that 2iKA = e^+ika/2x [(ik - k) (- ik + k)_e^+ka +(ik + k) (ik + k)_e^-ka]F
d. Using the hyperbolic trigonometric functions sinh x = e^x - e^-x/2 and cosh x = e^x + e^-x/2 and the relationship cosh^2 x - sinh^2 x = 1, show that |F/A|^2 = 16(kK)^2/16(kK)^2 + (4(k^2 - k^2)^2 + 16 (kK)^2) sinh^2 (ka) = 1/1 + [(k^2 + k^2 sinh^2(ka)]/4 (kk)^2
e. Plot the transmission probability for an electron as a function of energy for V_0 = 1.6 times 10^-19 J and a = 9.0 times 10^-10 m up to an energy of 8 times 10^19 J. At what energy is the tunneling probability 0.1? At what energy is the tunneling probability 0.02? f. Plot the transmission probability for an electron of energy 0.50 times 10^-19 J as a function of the barrier width for v_0 = 1.6 times 10^-19 J between 2 times 10^-10 and 8 times 10^-10 m. At what barrier width is the transmission probability 0.2?