A generalized scattering theory in quantum mechanics
- URL: http://arxiv.org/abs/2307.01235v1
- Date: Mon, 3 Jul 2023 11:07:46 GMT
- Title: A generalized scattering theory in quantum mechanics
- Authors: Huai-Yu Wang
- Abstract summary: In quantum mechanics textbooks, a single-particle scattering theory is introduced.
A generalized Lippmann-Schwinger scattering equation is derived.
The transition probability of the scattering of two free particles is identical to that of the reciprocal process.
- Score: 1.14219428942199
- License: http://creativecommons.org/publicdomain/zero/1.0/
- Abstract: In quantum mechanics textbooks, a single-particle scattering theory is
introduced. In the present work, a generalized scattering theory is presented,
which can be in principle applied to the scattering problems of arbitrary
number of particle. In laboratory frame, a generalized Lippmann-Schwinger
scattering equation is derived. We emphasized that the derivation is rigorous,
even for treating infinitesimals. No manual operation such as analytical
continuation is allowed. In the case that before scattering N particles are
plane waves and after the scattering they are new plane waves, the transition
amplitude and transition probability are given and the generalized S matrix is
presented. It is proved that the transition probability from a set of plane
waves to a new set of plane waves of the N particles equal to that of the
reciprocal process. The generalized theory is applied to the cases of one- and
two-particle scattering as two examples. When applied to single-particle
scattering problems, our generalized formalism degrades to that usually seen in
the literature. When our generalized theory is applied to two-particle
scattering problems, the formula of the transition probability of two-particle
collision is given. It is shown that the transition probability of the
scattering of two free particles is identical to that of the reciprocal
process. This transition probability and the identity are needed in deriving
Boltzmann transport equation in statistical mechanics. The case of identical
particles is also discussed.
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