English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The paper considers a slightly modified one-dimensional infinite mass-in-masschain. In the case of the long-wave approximation, which corresponds to thetransition to a continuous medium, we obtained a system of two equations, whichis a generalization of the classical mechanics Klein-Gordon-Fock equation andhas both optical and acoustic branches of the dispersion relation. Based onthis classical mechanics system of equations, we have proposed a system of tworelativistic quantum mechanics equations, which is a generalization of therelativistic quantum mechanics Klein-Gordon-Fock equation. Next, based on thissystem and following the Dirac approach, we have proposed the generalization ofthe Dirac equation for a free electron with an eight-component wave function inthe form of a system of eight linear partial differential equations of thefirst order. Unlike the Dirac equation with a four-component wave function,which has only an optical branch of the dispersion relation, the generalizedDirac equation has both optical and acoustic branches of the dispersionrelation, each of which has two branches with positive and negative energies,respectively. We have calculated phase and group velocities for all cases. Forthe positive and negative acoustic branches, the phase and group velocities areequal in modulus to the speed of light. For the positive and negative opticalbranches, the phase and group velocities have a structure like that of deBroglie waves. In the one-dimensional case, eight linearly independentsolutions corresponding to eight combinations of two branches of dispersion,two signs of total energy, and two possible directions of spin orientation,each in the form of four plane waves, are obtained.

A Mass-in-Mass Chain and the Generalization of the Dirac Equation with  an Eight-Component Wave Function and with Optical and Acoustic Branches of  the Dispersion Relation