### Positive and Negative Particle Masses in the Bicubic Equation Limiting Particle Velocity Formalism

#### Abstract

The interest in the negative particle mass here got encouraged by the Rachel Gaal July 2017 APS article (Gaal, 2017)

describing Khamehchi et al. (2007) observation of an effective negative mass in a spin-orbit coupled Bose-Einstein

condensate. Hence, since in the bicubic equation limiting particle velocity formalism (Soln, 2014, 2015, 2016, 2017)

positive m+ = m ≻ 0 and negative m− = −m ≺ 0 masses with m2+ = m2− = m2 are equally acceptable, then from a purely

theoretical point of view, the evaluation of particle limiting velocities for both m+ and a m− masses should be done.

Starting with the original solutions for particle limiting velocities c1; c2 and c3, given basically for a positive particle

mass m+ (Soln, 2014, 2015, 2016, 2017), now also are done for a negative particle mass m− This is done consistent with

the bicubic equation mathematics, by solving for c1; c2 and c3 not only form+ but also for m−. Hence, in addition to

having the limiting velocities of positive mass m+ primary, obscure and normal particles, now one has also the limiting

velocities of negative mass m− primary, obscure and normal particles, however, numerically equal to limiting velocities,

respectively of m+ masses obscure, primary and normal particles, forming the m+ and m− masses of equal limiting velocity

value doublets : c1(m−) = c2(m+), c2(m−) = c1(m+) , c3(m−) = c3(m+). Now, one would like to know as to which particle

with a negative mass m− = −m ≺ 0, obtained from the positive mass m+ = m ≻ 0 with the substitution m − −m, can

have a real limiting velocity? It turns out that it is the obscure particle limiting velocity c2(m+) that changes from the

imaginary value, c22(m+) ≺ 0, into the real limiting velocity value c22(m−) ≻ 0 when the change m+ − m− is made and,

at the same time, retaining the same energy. Similar procedure applied to the original primary particle limiting velocity

starting with c21(m+) ≻ 0 , keeping the total energy the same,with the change m − −m one ends up with c21

(m−) ≺ 0 that is, imaginary c1. The procedure of changing m+ − m− in normal particle limiting velocity causes no change, it remains the same realc3. Because m2 (= m2+ = m2−), E2 and v2 remain the same , these mass regenerations, m+ − m− and m− − m+ could in principle also occur spontaneously.

describing Khamehchi et al. (2007) observation of an effective negative mass in a spin-orbit coupled Bose-Einstein

condensate. Hence, since in the bicubic equation limiting particle velocity formalism (Soln, 2014, 2015, 2016, 2017)

positive m+ = m ≻ 0 and negative m− = −m ≺ 0 masses with m2+ = m2− = m2 are equally acceptable, then from a purely

theoretical point of view, the evaluation of particle limiting velocities for both m+ and a m− masses should be done.

Starting with the original solutions for particle limiting velocities c1; c2 and c3, given basically for a positive particle

mass m+ (Soln, 2014, 2015, 2016, 2017), now also are done for a negative particle mass m− This is done consistent with

the bicubic equation mathematics, by solving for c1; c2 and c3 not only form+ but also for m−. Hence, in addition to

having the limiting velocities of positive mass m+ primary, obscure and normal particles, now one has also the limiting

velocities of negative mass m− primary, obscure and normal particles, however, numerically equal to limiting velocities,

respectively of m+ masses obscure, primary and normal particles, forming the m+ and m− masses of equal limiting velocity

value doublets : c1(m−) = c2(m+), c2(m−) = c1(m+) , c3(m−) = c3(m+). Now, one would like to know as to which particle

with a negative mass m− = −m ≺ 0, obtained from the positive mass m+ = m ≻ 0 with the substitution m − −m, can

have a real limiting velocity? It turns out that it is the obscure particle limiting velocity c2(m+) that changes from the

imaginary value, c22(m+) ≺ 0, into the real limiting velocity value c22(m−) ≻ 0 when the change m+ − m− is made and,

at the same time, retaining the same energy. Similar procedure applied to the original primary particle limiting velocity

starting with c21(m+) ≻ 0 , keeping the total energy the same,with the change m − −m one ends up with c21

(m−) ≺ 0 that is, imaginary c1. The procedure of changing m+ − m− in normal particle limiting velocity causes no change, it remains the same realc3. Because m2 (= m2+ = m2−), E2 and v2 remain the same , these mass regenerations, m+ − m− and m− − m+ could in principle also occur spontaneously.

#### Full Text:

PDFDOI: https://doi.org/10.5539/apr.v10n1p14

Copyright (c) 2018 Josip Soln

License URL: http://creativecommons.org/licenses/by/4.0

Applied Physics Research ISSN 1916-9639 (Print) ISSN 1916-9647 (Online) Email: apr@ccsenet.org

Copyright © Canadian Center of Science and Education

To make sure that you can receive messages from us, please add the 'ccsenet.org' domain to your e-mail 'safe list'. If you do not receive e-mail in your 'inbox', check your 'bulk mail' or 'junk mail' folders.