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I was looking at solved example (3.13) in the Schaum's Series book on QM by Yoav Peleg et al (2nd edn), where they solve for a step potential where a high energy particle is coming from the left and going right (the step potential's rise being from left to right). In the explanation, the author says that

"since the particle is known to go from left to right, hence we can omit the left-travelling wave in the high potential (right side) region)".

I am uncomfortable with this kind of justification, my discomfort originating from the fact that in QM I have seen that sometimes we resort to intuitive analogues as in the above example, but sometimes we don't and we're encouraged not to because QM is often counter-intuitive (eg. in a periodic crystal, the momentum of the Bloch electron is different from the crystal momentum $\hbar k$

Is there a better or a more formal justification for the above example as regards why we omit the left travelling wave in the high potential region?

Qmechanic
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vharihar
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1 Answers1

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The hidden point here is that we are solving not an eigenvalue problem, as a harmonic oscillator or an infinite square well, but a scattering problem. The (here Schrödinger) equation for the two types of problems is the same, but the boundary conditions are different. In the eigenvalue problems we typically impose normalization of the total probability and depand that the wave function vanishes at infinity. In the scattering problems we normalize by the particle flux. There are different ways to justify this choice: e.g., this can be related to the boundary conditions in time (at $t=-\infty$). We can also solve an equivalent problem with outgoing boundary conditions - and it can be shown that the incoming and outgoing solutions are two linearly independent solutions for each energy.

These issues are properly dealt (i.e., without resorting to intuition) when studying scattering theory - which usually takes place towards the end of the quantum mechanics course. These questions are also typically revised in the context of quantum field theory, where calculating scattering cross-sections is one of the central subjects. Introducing them in the very beginning, would unnecessarily complicate the things... but unfortunately results in many (completely justified) questions regarding such simple problems as scattering from a potential step.

Remark see also my answers here and here for a bit more details.

Roger V.
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