A computer can solve several equations in several unknowns easily, and here we study a particular E> U0 barrier problem, where all the values it needs are real. Once the computer finds the multiplicative constants of all the functions involved, we can verify equations (6-13) as well as see what is happening when the particles are “over the barrier.” Still, it helps to simplify things as much as possible. With length, time, and mass at our disposal, we can choose our units so that the particle mass and the value of ft are both 1 and the barrier width L is exactly n. Suppose that in this system of units, the energy E of the incident particles is 1.125, and the barrier height Un is I. Furthermore, because only ratios are ever really needed, assume the multiplicative constant A for the incident wave function is 1. (a) Write down solutions to the Schrödinger equation in the three regions, using numerical values wherever possible, then write down the smoothness conditions, (b) You should have four equations in four unknowns. (They should be real—don't forget the Euler formulas.) Use a computer to solve for the unknowns, (c) Do the reflection and transmission probabilities given by your results agree with the general formulas given in (6-13)? (d) If a particle were located in the region "over the barrier" between x = 0 and x = L, is it just as likely to be a particle moving left as right? If not, which is more probable, and why?
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