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De Broglie waves meet Schrödinger's equation

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For a quantum-mechanical free particle, de Broglie's hypothesis relates group velocity vg to phase velocity vp via vgvp  = c 2. However, the relation has not yet been grounded in the usual definition vg  = dω/dk. This article provides the missing dispersion relation to evaluate dω/dk. Direct solution of Schrödinger's equation in one dimension yields the dispersion relation ω = (ħ/2 m)k 2, from which follows the relation vg  = 2vp . Further implications of the dispersion relation are explored. Because the free-particle Schrödinger equation is a heat equation with an imaginary heat constant, its Green's function is a linear chirp that fills all space starting the instant after a delta-function initial condition. Particle localization is possible only when the Green's function is spatially convolved (at a time t) with a substantially band-limited initial wavefunction. The faster expansion of a wavefunction when it is initially narrow could be a realization of the uncertainty principle. All of this Schrödinger-equation analysis, however, makes a disconcerting break with relativistic quantum mechanics. Free-particle solution of the Klein‐Gordon equation vindicates vgvp  = c 2. The nonrelativistic noncorrespondence of Schrödinger and Klein‐Gordon, or more simply of vg  = 2vp with vgvp  = c 2, leads us to ask which can be trusted in the nonrelativistic velocity limit.
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Keywords: Free Particle; Green’s Function; Group Velocity; Quantum Mechanics; Schrödinger Equation, Phase Velocity

Document Type: Research Article

Publication date: December 30, 2013

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  • Physics Essays has been established as an international journal dedicated to theoretical and experimental aspects of fundamental problems in Physics and, generally, to the advancement of basic knowledge of Physics. The Journal's mandate is to publish rigorous and methodological examinations of past, current, and advanced concepts, methods and results in physics research. Physics Essays dedicates itself to the publication of stimulating exploratory, and original papers in a variety of physics disciplines, such as spectroscopy, quantum mechanics, particle physics, electromagnetic theory, astrophysics, space physics, mathematical methods in physics, plasma physics, philosophical aspects of physics, chemical physics, and relativity.
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