A computer makes solving several equations in several unknowns pretty easy, so provided your computer can handle complex numerical values, finding the multiplicative constants of all the functions in a tunneling problem isn't too bad. Not only can we verify equations (6-16), but we can also see what the functions inside the barrier are doing. 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 m and the value of % arc both 1 and the barrier the energy E of the incident particles is 2, and the barrier height U0 is 4, Furthermore, because only ratios are ever really needed, assume the multiplicative constant A for the incident wave function is I. (a) Write down solutions to the Schrödinger equation in the three regions, using numeiical values wherever possible, Then, write down the smoothness conditions, (b) You should have four equations in four unknowns, with some of the known constants being complex. Use a computer to solve for the unknowns, (c) Do the reflection and b'ansmission probabilities given by your results agree with the general formulas given in (6.16)? (d) In Section 6.2, it is said that the coefficient of the exponentially increasing function inside the barrier is usually small, implying that the function inside the barrier is essentially a decaying exponential. Do your results agree?
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