Introduction to Quantum Mechanics 2

Appendix 3Transparency of a Potential Barrier, Resonance

A3.1. Expression of transparency

Physical phenomena related to the transparency of a rectangular potential barrier of height V₀ and width a have been studied in Chapter 1. We have studied the particular case of a particle of total energy E < V₀ that moves toward the barrier from a point of abscissa x < 0. The profile of the barrier is schematically represented in Figure 1.7. This appendix completes the description of the behavior of the particle in contact with the barrier and analyzes Schrödinger’s equation in the three zones I, II and III for E >V₀.

If Schrödinger’s stationary equation is applied to zones I, II and III, expressions [1.102], [1.103] and [1.104] are obtained as follows:

[A3.1]

We put:

[A3.2]

Solutions to equations [A3.1] can be written as follows:

[A3.3]

Let us now express the boundary conditions in x = 0, and then in x = a. Using [A3.3], we get:

[A3.4]

[A3.5]

The transparency of the barrier is T = |F|²