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The Relation and Transformation between Particle Characteristics and Wave Characteristics of Objects—Relativistic Study of the Second-Order Partial Differential Equation of Mass, Kinetic Energy and Force to Space-Time
DOI: 10.12677/MP.2023.135013, PDF, HTML, XML, 下载: 337  浏览: 658

Abstract: The second-order partial differential equation of mass, kinetic energy and force with respect to time-space derived from classical mechanics, , namely MEF equation, shows the instantaneous effects of mass, kinetic energy and force in time-space and reflects the particle characteristics of objects. When relativistic effect is considered, both physical particle and photon have corresponding relativistic MEF equations. The physical particle’s relativistic MEF equation can be decomposed into particle-characteristic and wave-characteristic MEF equations, reflecting the wave-particle duality. If it reaches the speed of light, the physical particle’s relativistic MEF equation can be completely transformed into the relativistic MEF equation for a photon, and the physical particle will be transformed into a photon. When it nears or reaches the speed of light, physical particle’s motion mass should be calculated by the formula containing the motion mass of photons. When a physical particle is accelerated, it is possible to be accelerated to the speed of light and transformed into a photon.

1. 引言

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}=m\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (1)

MEF方程揭示了作用力和质量的本质以及质量和能量相互转化的时空原理，反映了质量、动能和作用力之间的内在关系，以及它们在时空中的瞬间效应。当不考虑相对论效应时，MEF方程主要揭示了物体的粒子特性，没有涉及到物体的波动特性。

2. 相对论与非相对论MEF方程

$m=\frac{{m}_{0}}{\sqrt{1-{v}^{2}/{c}^{2}}}$ (v为粒子速度，c为光速)(2)

$v\ll c$ 时，不考虑相对论效应，式(1)中物体的运动质量m可认为是不变量，和物体静止质量m0相等，这时MEF方程可表述为

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}={m}_{0}\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (m0为静止质量) (3)

3. 光子的相对论MEF方程

$E=h\gamma =m{c}^{2}$ (h为普朗克常量，γ为光子频率) (4)

$m=\frac{E}{{c}^{2}}$ (5)

${|\frac{\partial E}{\partial l}|}^{2}=\frac{E}{{c}^{2}}\frac{{\partial }^{2}E}{\partial {t}^{2}}$ (6)

3.1. 光子的波动特性

$E\left(l,t\right)=A{\text{e}}^{i\left(k\cdot }{{}^{l}}^{-\omega t\right)}$ (其中 $\omega /k=c$ ) (7)

3.2. 光子的粒子特性

$\frac{\partial E}{\partial l}=F={\left(\frac{E}{{c}^{2}}\frac{{\partial }^{2}E}{\partial {t}^{2}}\right)}^{1/2}\frac{\text{d}l}{\text{d}l}$ (8)

4. 实物粒子的相对论MEF方程

${m}_{0}=m\sqrt{1-{v}^{2}/{c}^{2}}=m\sqrt{1-{\beta }^{2}}$ (9)

${E}_{k}=m{c}^{2}-{m}_{0}{c}^{2}$ (10)

${E}_{k}=m{c}^{2}-{m}_{0}{c}^{2}=\left(1-\sqrt{1-{\beta }^{2}}\right)m{c}^{2}=\left(1-\sqrt{1-{\beta }^{2}}\right)\frac{{v}^{2}}{{\beta }^{2}}m=\frac{1}{1+\sqrt{1-{\beta }^{2}}}m{v}^{2}$ (11)

$m=\left(1+\sqrt{1-{\beta }^{2}}\right)\frac{{E}_{k}}{{v}^{2}}$ (12)

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}=\left(1+\sqrt{1-{\beta }^{2}}\right)\frac{{E}_{k}}{{v}^{2}}\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (其中 $\beta =v/c$ ) (13)

4.1. 实物粒子的粒子特性

$v\ll c$ 时，即 $\beta \approx 0$

$1+\sqrt{1-{\beta }^{2}}\approx 2$ (14)

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}=2\frac{{E}_{k}}{{v}^{2}}\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (15)

$\beta \approx 0$ 及式(9)、式(11)可得

${E}_{k}=\frac{1}{2}m{v}^{2}=\frac{1}{2}{m}_{0}{v}^{2}$ (16)

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}={m}_{0}\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (3)

4.2. 实物粒子的波动特性

$1+\sqrt{1-{\beta }^{2}}\approx 1$ (17)

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}=\frac{{E}_{k}}{{v}^{2}}\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (18)

${E}_{k}\left(l,t\right)=A{\text{e}}^{i\left(k\cdot }{{}^{l}}^{-\omega t\right)}$ (其中 $\omega /k=v$ ) (19)

$\psi \left(x,t\right)=A{\text{e}}^{i\left(p\cdot x-\epsilon t\right)/\hslash }=A{\text{e}}^{i\left(k\cdot x-\omega t\right)}$ (20)

4.3. 实物粒子的波粒二象性

$\sqrt{1-{\beta }^{2}}{|\frac{\partial {E}_{k}}{\partial l}|}^{2}=2\sqrt{1-{\beta }^{2}}\frac{{E}_{k}}{{v}^{2}}\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (21)

$\left(1-\sqrt{1-{\beta }^{2}}\right){|\frac{\partial {E}_{k}}{\partial l}|}^{2}=\left(1-\sqrt{1-{\beta }^{2}}\right)\frac{{E}_{k}}{{v}^{2}}\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (22)

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}=2\frac{{E}_{k}}{{v}^{2}}\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (15)

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}=\frac{{E}_{k}}{{v}^{2}}\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (18)

Table 1. The values of particle characteristic coefficient 1 − β 2 and the wave characteristic coefficient ( 1 − 1 − β 2 ) of physical particles corresponding to different β (= v/c) values

Figure 1. The proportions of particle characteristic and wave characteristic of physical particles vary with β

5. 实物粒子与光速

$m=\frac{{m}_{0}}{\sqrt{1-{v}^{2}/{c}^{2}}}$ (2)

${m}_{p}=m\left(1-\sqrt{1-{\beta }^{2}}\right)$ (23)

$m=\frac{{m}_{p}}{1-\sqrt{1-{v}^{2}/{c}^{2}}}$ (24)

6. 结论

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