Forward-scattering off a potential well

Question

In his book The Physics of Quantum Mechanics, James Binney writes the following:

The scattering cross-section. In the case that $V_0<0$, so the scattering potential forms a potential well, the outgoing wave at $x>a$ represents two physically distinct possibilities: (i) that the incoming particle failed to interact with the potential well and continued on its way undisturbed, and (ii) that it was for a while trapped by the well and later broke free towards the right rather than the left. We isolate the possibility of scattering by writing the amplitude of the outgoing wave as $1+T$ times the amplitude of the incoming wave. Here one represents the possibility of passing through undisturbed and $T$ represents real forward scattering.

My question is what is the meaning of this expansion as $1+T$? I say this because you don't tend to consider the possible "paths" that a particle takes when solving simple eigenstate problems like this. The only other place where we talk about such a thing I can think of is Feynman path integral kind of things. Is this interpretation related to that?

Answer 1

As is often true, it is useful to think of limiting cases. In the limit where the scattering potential goes to zero, the wave continues on its way completely undisturbed by the potential, and thus the outgoing amplitude is just $1$ times the incoming amplitude. So for the case where the potential is non-zero, splitting into $1 + T$ separates what you would have gotten with no scattering potential from what you get only because of the non-zero scattering potential.

In other words, writing the outgoing amplitude as $1 + T$ (times the incoming amplitude) lets $T$ (times the incoming amplitude) represent the difference in the outgoing amplitude due to the presence of the scattering potential. In the limit where the potential goes to zero, $T$ also goes to zero.

Answer 2

What is the problem? Binney's explanation is perfectly clear. If |B| is the amplitude of the outgoing wavefunction, you just write it as $|B|=(1+T)|A|=|A| + T|A| =$ wave which past undisturbed + wave which was changed through interaction by a transmission probability T. You are not summing the possible paths, you are just writing dividing a general outgoing wavefunction into two parts.