Question

A particle moving in a region of zero force encounters a precipice---a sudden drop in the potential energy to an arbitrarily large negative value. What is the probability that it will “go over the edge”?

Short answer

I will definitely reflect for $U_0$tending to infinite.

Step-by-step solution

Given known parameters

$T=4\frac{\sqrt{E(E-U_0})}{{\left(\sqrt{E+\sqrt{E-U_0}}\right)}^2}$

Solution

As $U_0$tends to $\infty$, the transmission coefficient tends to $0$

Thus reflection is definitely possible.

Explanation and Conclusion

We might suppose that there should be a smooth match to an answer within the region beyond the downward step. But the actual fact that the P.E. is infinite means the derivative is discontinuous at the drop. The wave function is solely an undulation to the left of the step, zero at the step, and nil beyond the step.