Time-dependent theory of scattering a particle on a one-dimensional $\delta$-potential

• URL: http://arxiv.org/abs/2305.15381v4
• Date: Fri, 11 Aug 2023 04:09:51 GMT
• Title: Time-dependent theory of scattering a particle on a one-dimensional $\delta$-potential
• Authors: N. L. Chuprikov
• Abstract summary: Modern time-dependent quantum theory of scattering a particle on a one-dimensional $delta$-potential is internally inconsistent. One-particle states should be identified with vector functionals on the algebra of operators, and not with vectors or rays in Hilbert space.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: It is shown that the modern time-dependent quantum theory of scattering a particle on a one-dimensional $\delta$-potential is internally inconsistent and gives erroneous results: the asymptotic completeness is not really satisfied in this process, and the scattering states themselves are not pure. The asymptotic superselection rule applies here, according to which the space of asymptotes of scattering states is the direct sum of two coherent sectors associated with different signs of the particle momentum at $t\to\mp\infty$: any superposition of asymptotes from different sectors is a mixed state. The Schr\"{o}dinger dynamics crosses the boundaries of these coherent sectors. So the two-sided scattering process is a mixture of two one-sided ones. In turn, each of the latter is a mixture of transmission and reflection subprocesses, which are described by the Schr\"{o}dinger dynamics only at $t\to\mp\infty$. Our approach shows that one-particle states should be identified with vector functionals on the algebra of unbounded operators, and not with vectors or rays in Hilbert space (which should be considered as auxiliary concepts in quantum mechanics).

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