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On a Unitary Solution for the Schrödinger Equation
Version 1
: Received: 20 September 2024 / Approved: 23 September 2024 / Online: 23 September 2024 (15:19:38 CEST)
How to cite: Mulian, A. Y. On a Unitary Solution for the Schrödinger Equation. Preprints 2024, 2024091809. https://doi.org/10.20944/preprints202409.1809.v1 Mulian, A. Y. On a Unitary Solution for the Schrödinger Equation. Preprints 2024, 2024091809. https://doi.org/10.20944/preprints202409.1809.v1
Abstract
For almost 75 years, the general solution for the Schrödinger equation was assumed
to be generated by an exponential or a time-ordered exponential known as the Dyson
series. We study the unitarity of this solution in case of singular Hamiltonian and provide a
new methodology that is not based on the assumption that the underlying space is L2(R).
Then, an alternative operator for generating the time evolution is suggested that is manifestly
unitary, regardless of the choice of the Hamiltonian. The new construction involves an additional
positive operator that normalizes the wave-function locally and allows us to preserve
unitary even on indefinite norm spaces. Our considerations show that Schrödinger's and Liouville's
equations are, in fact, two sides of the same coin, and together they become the unified
description of quantum systems.
Keywords
Quantum mechanics; Schrodinger equation; Unitarity
Subject
Physical Sciences, Particle and Field Physics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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