Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.35.
The problem of the finite step potential can be inverted to give a finite drop potential by replacing by , so the potential is given by
If we assume particles coming in from the left, then we must have (otherwise the wave function would decay exponentially inside the barrier and we couldn’t have particles coming in from infinity on the left). In this case the reflection coefficient is
and the transmission coefficient is
For the problem reduces to that of the free particle, and , as we’d expect. As gets very large, , .
If we take , then .
Although the graph of the potential looks like a cliff, it doesn’t represent the behaviour of an object, such as a car, falling over a cliff. Classically, the energy of a car is kinetic + potential, which in the absence of other forces, remains a constant. If a car had a speed in a region where , then its total energy is kinetic: . If it suddenly encounters a region where , then we’d have , so is larger than , meaning that the car would instantaneously increase its speed, which of course doesn’t happen. In reality, a car driving off a cliff encounters a potential energy of where is the distance it has fallen, so its kinetic energy increases gradually. Besides, a car falling off a cliff is essentially a two-dimensional problem, so trying to analyze it in one dimension won’t work.
A slightly more realistic case is that of a neutron which is fired at an atomic nucleus. The neutron experiences a sudden drop in potential from outside the nucleus to MeV inside. If we give the neutron an initial kinetic energy of MeV, then the probability of transmission into the nucleus is