Because the potential energy within each region is a constant, you already know the form of the solution to Schrödinger's Equation within each one, though not yet the values of the coefficients of each of the functions. Within any region with constant potential, in general there are two independent functions, so in this problem, use A and B for the coefficients of those functions in region I, use C and D in region II, while in region III you should recognize that the boundary condition at +infty eliminates one of its two functions, so there is only one coefficient remaining, F.
(a) Write the general solutions for psi _(1)(x),psi _(2)(x), and psi _(3)(x) in regions I, II, and III respectively, using the coefficients A,B,C,D, and F as described above, and explain how you came to each conclusion. You can use the shorthand k_(1),k_(2), and k_(3) for each corresponding wavevector or decay constant, but you must explicitly define each of those values in terms of quantities given in the statement of the problem and fundamental constants, and explain how you came to each of those results.
(b) Write the two continuity equations that apply at each boundary. Then write the four equations that result from the application of the continuity conditions in terms of the coefficients A,B,C,D, and F and k_(1),k_(2), and k_(3). Note that among the set of 8 symbols in the previous sentence, there are only 4 unknowns, so these 4 equations are sufficient to solve the entire problem.
Important Note #1: Do not solve for the values of the coefficients. You are only being asked to write four equations which could be solved. However, solving them is merely some tedious algebra that won't teach you anything useful for future problems.
Important Note #2: You will be graded on your explanation of why you wrote each of your answers. No credit will be given to any answer merely because the equation was correct.
Because the potential energy within each region is a constant, you already know the form of the solution to Schrodinger's Equation within each one, though not yet the values of the coefficients of each of the functions. Within any region with constant potential, in general there are two independent functions, so in this problem, use A and B for the coefficients of those functions in region I, use C and D in region II, while in region III you should recognize that the boundary condition at +oo eliminates one of its two functions, so there is only one coefficient remaining, F.
(a) Write the general solutions for b1(), $2(), and /3() in regions I, II, and III respec tively, using the coefficients A, B, C, D, and F as described above, and erplain how you came to each conclusion. You can use the shorthand ki, k2, and ks for each cor- responding wavevector or decay constant, but you must explicitly define each of those values in terms of quantities given in the statement of the problem and fundamental constants, and explain how you came to each of those results.
(b) Write the two continuity equations that apply at each boundary. Then write the four equations that result from the application of the continuity conditions in terms of the coefficients A, B, C, D, and F and ki, k2, and k3. Note that among the set of 8 symbols in the previous sentence, there are only 4 unknowns, so these 4 equations are sufficient to solve the entire problem.
Important Note #l: Do not solve for the values of the coefficients. You are only being asked to write four equations which could be solued. However, solving them is merely some tedious algebra that won't teach you anything useful for future problems. Important Note #2: You will be graded on your explanation of why you wrote each of your answers. No credit will be given to any answer merely because the equation was correct.
