Why is the reflection coefficient in quantum mechanical scattering defined this way?

Question

In Griffiths' "Introduction to Quantum Mechanics, second edition" section 2.5.2, p. 73, he states: For the delta-function potential, when considering the scattered states (with $E > 0$), we have the general solutions for the time-independent Schrodinger equation:

$$\psi(x) = Ae^{ikx} + Be^{-ikx}\quad\text{for}\quad x<0 \tag{2.131}$$

and

$$\psi(x) = Fe^{ikx} + Ge^{-ikx}\quad\text{for}\quad x>0.\tag{2.132} $$

In a typical scattering experiment, particles are fired in from one direction-let's say, from the left. In that case the amplitude of the wave coming in from the right will be zero: $$G=0\quad(\text{for scattering from the left}).\tag{2.136}$$

Then $A$ is the amplitude of the incident wave, $B$ is the amplitude of the reflected wave and $F$ is the amplitude of the transmitted wave. Now the probability of finding the particle at a specified location is given by $|\Psi|^2,$ so the relative probability that an incident particle will be reflected back is $$R \equiv \frac{|B|^2}{|A|^2},\tag{2.138}$$ where $R$ is called the reflective coefficient.

Question:

How does the definition of $R$ follow? Where exactly does this probability come from?

Answer 1

Sketched proof:

1. The first question the reader should ask him/herself is:

   Why can we use the time-independent Schrödinger equation to describe scattering of an incoming particle against a fixed potential, which naively sounds like a time-dependent process?

   This is answered in e.g. this Phys.SE post. In particular, note that $e^{+ikx}$ and $e^{-ikx}$ are a right- and left-mover, respectively.

2. Secondly, to conserve probability over time, impose that the $S$-matrix should be unitary. This is e.g. done in my Phys.SE answer here. Unitarity implies with $G=0$ that $$|A|^2~=~|B|^2+|F|^2.$$

3. Thirdly, interpret $\frac{|B|^2}{|A|^2}$ and $\frac{|F|^2}{|A|^2}$ as probabilities for reflection and transmission, respectively, thereby adding up to 100 %.