﻿ 基于相对论的粒子波函数与实验分析

# 基于相对论的粒子波函数与实验分析Particle Wave Function and Experimental Analysis Based on Relativity Theory

Abstract: The Schrödinger equation uses the Planck constant as a characteristic quantity of the quantum mechanical system and is not easily solved in practice. The purpose of this paper is to establish a particle wave function equation based on the theory of relativity, which can be directly used to calculate particle experimental data and describe the physical reality of particle motion. This paper expounds the wave-particle duality of particle, introduces the particle’s rest mass constant into Einstein’s mass-energy equation of moving particles, introduces the particles moving mass into Newtonian kinetic energy equation, and the relativistic momentum operator , energy operator and Hamiltonian operator are proposed, and the Schrödinger equation is modified to be the relativistic particle wave function equation. In this paper, photon double-slit shooting experiments show that photons fluctuating in the lattice of air molecules, change their refractive index or reflectivity through the double-slit asymmetric energy field, and test proves that the position vector in the relativistic particle wave function equation is the particle fluctuating path vector. Through experimental data of hydrogen atom radiation, the electron wave function and probability wave of hydrogen atom are tested, and that the statistical explanation based on is in line with the physical reality of electron motion of hydrogen atoms is proved. The relativistic particle wave function will play an important role in the development and application of quantum physics.

1. 引言

1900年普朗克提出能量常数h假设，1905年爱因斯坦将普朗克常数h作为光子的能量常数，1924年，德布罗意 [1] 提出物质波假设，将普朗克常数h引入粒子或物质波计算公式。德布罗意认为光子的动量 $p=h/\lambda$ ，光子的波长 $\lambda =h/p$ ，并推导出物质波波长，或 $\lambda =h/\sqrt{2mE}$ ，其中m、E为实物粒子的质量、能量；实物粒子的频率 $\nu =E/h$ ，角频率 $\omega =E/\hslash$ ，波矢 $k=p/\hslash$ ，其中 $\hslash =h/2\text{π}$。依据德布罗意假设，自由粒子平面波函数

$\psi \left(r,t\right)=A\mathrm{exp}\frac{i}{\hslash }\left(p\cdot r-Et\right),$ (1)

1926年，薛定谔 [3] [4] [5] [6] 将德布罗意物质波引入波函数理论，提出基于经典理论的波函数方程，描述粒子的波动性、运动状态和量子化性质，揭示粒子运动的基本规律。如一般情况下粒子微分运动薛定谔方程

$i\hslash \frac{\partial }{\partial t}\psi \left(r,t\right)=\left[-\frac{{\hslash }^{2}}{2m}{\nabla }^{2}+V\left(r,t\right)\right]\psi \left(r,t\right),$ (2)

$\left[-\frac{{\hslash }^{2}}{2m}{\nabla }^{2}+V\left(r\right)\right]\psi \left(r\right)=E\psi \left(r\right),$ (3)

1926年，玻恩 [7] [8] 对薛定谔波函数进行统计诠释，认为薛定谔波函数是粒子几率波函数，论述了薛定谔波函数的合理性。

2. 粒子波函数的相关概念

2.1. 基本粒子波长与外场能量密度波长

2.2. 基本粒子波动与外场势能波动

2.3. 基于相对论的粒子波长与动量

1900年，普朗克提出的能量常数h，实际上是电子的能量常数。普朗克常数h与电子静止质量 ${m}_{e}$、电子能量E、康普顿电子波长 ${\lambda }_{c}$ 之间的关系， $h={m}_{e}c{\lambda }_{c}$$E=hc/{\lambda }_{c}$${\lambda }_{c}=h/{m}_{e}c$。1905年，爱因斯坦提出质能关系式 $E=m{c}^{2}$ ，电子的静止质量转换成能量 ${m}_{e}{c}^{2}=hc/{\lambda }_{c}$ ，所以 ${\lambda }_{c}=h/{m}_{e}c$${m}_{e}=h/c{\lambda }_{c}$。康普顿电子波长 ${\lambda }_{c}$ 是计算电子静止质量的波长。因此，依据基本粒子的波长(如各种光子的波长)计算其静止质量是合理的。粒子的质量也是量子化的，用粒子静止质量常数 ${h}_{m}=\text{5}\text{.45025577174353}×{\text{10}}^{-65}\text{\hspace{0.17em}}\text{kg}\cdot {\text{m}}^{3}$ [22] 和粒子波长计算粒子静止质量

$m=\frac{1}{\frac{4}{3}\text{π}{\lambda }^{3}}{h}_{m},$ (4)

$m=\frac{1}{\frac{4}{3}\text{π}{\left(\alpha \lambda /2\text{π}\right)}^{3}}{h}_{m}=\frac{6{\text{π}}^{2}}{{\alpha }^{3}{\lambda }^{3}}{h}_{m},$ (5)

${v}_{e}=\alpha c/\sqrt{\frac{{\lambda }_{e}}{{\lambda }_{0}}},$ (6)

${\lambda }_{0}=\frac{{\lambda }_{e}}{{\left(\alpha c/{v}_{e}\right)}^{2}}$${\lambda }_{0}=k{\lambda }_{c},$ (7)

${\lambda }_{e}={\left(\alpha c/{v}_{e}\right)}^{2}{\lambda }_{0}.$ (8)

$E=\frac{hc}{{\lambda }_{e}}.$ (9)

${m}_{m}=m/\sqrt{1-\left({v}^{2}/{c}^{2}\right)}-m,$ (10)

$m+{m}_{m}=m/\sqrt{1-\left({v}^{2}/{c}^{2}\right)}.$ (11)

$E={m}_{m}{c}^{2}.$ (12)

$E=\left(\frac{1}{2}m+{m}_{m}\right){v}^{2},$ (13)

${m}_{m}{c}^{2}=r\left(\frac{1}{2}m+{m}_{m}\right){v}^{2}.$ (14)

$E=\frac{1}{2}\left(m+{m}_{m}\right){v}^{2},$ (15)

${m}_{m}{c}^{2}=r\frac{1}{2}\left(m+{m}_{m}\right){v}^{2}.$ (16)

$p=\left(m+{m}_{m}\right)v=\sqrt{\left(2m+{m}_{m}\right)E},$ (17)

3. 基于相对论的粒子波函数

3.1. 基于相对论的动量、能量与哈密顿量算符

Table 1. Data table of particle rest mass, Particle wavelength, Particle wavelength radius, velocity, energy, momentum, Particle motion wavelength and de Broglie wavelength

$p=\frac{6{\text{π}}^{2}v{h}_{m}}{{\alpha }^{3}{\lambda }_{c}^{3}\sqrt{1-\left({v}^{2}/{c}^{2}\right)}},$ (18)

$k=\frac{4\text{π}}{\alpha \lambda }$ , $\omega =\frac{2\text{π}c}{\lambda }.$ (19)

$\beta =\frac{p}{k{h}_{m}}=\frac{3\text{π}v\lambda }{2{\alpha }^{2}{\lambda }_{c}^{3}\sqrt{1-\left({v}^{2}/{c}^{2}\right)}},$ (20)

$\stackrel{^}{p}\equiv i\beta {h}_{m}\nabla$ , $\stackrel{^}{E}=i\beta {h}_{m}\frac{\partial }{\partial t}$$\stackrel{^}{H}=-\frac{{\beta }^{2}{h}_{m}^{2}}{2m+{m}_{m}}{\nabla }^{2}+V\left(r,t\right),$ (21)

3.2. 基于相对论的粒子波函数方程

$\psi \left(r,t\right)=A\mathrm{exp}\frac{i}{\beta {h}_{m}}\left(p\cdot r-Et\right),$ (22)

${\psi }_{p}\left(r\right)=\frac{1}{{\left(2\text{π}\beta {h}_{m}\right)}^{3/2}}{\text{e}}^{\frac{i}{\beta {h}_{m}}\left(p\cdot r\right)},$ (23)

$i\beta {h}_{m}\frac{\partial }{\partial t}\psi \left(r,t\right)=\left[-\frac{{\beta }^{2}{h}_{m}^{2}}{2m+{m}_{m}}{\nabla }^{2}+V\left(r,t\right)\right]\psi \left(r,t\right),$ (24)

$\left[-\frac{{\beta }^{2}{h}_{m}^{2}}{2m+{m}_{m}}{\nabla }^{2}+V\left(r\right)\right]\psi \left(r\right)=E\psi \left(r\right),$ (25)

4. 杨氏双缝实验对波函数的检验

4.1. 双缝的折射与反射原理

4.2. 粒子的路径波与波函数方程

4.3. 量子态迭加原理

$\psi \left(r\right)=\frac{1}{{\left(2\text{π}\beta {h}_{m}\right)}^{3/2}}\int \phi \left(p\right){\text{e}}^{\frac{i}{\beta {h}_{m}}\left(p\cdot r\right)}{\text{d}}^{3}p,$ (26)

$\phi \left(p\right)=\frac{1}{{\left(2\text{π}\beta {h}_{m}\right)}^{3/2}}\int \phi \left(r\right){\text{e}}^{\frac{i}{\beta {h}_{m}}\left(p\cdot r\right)}{\text{d}}^{3}x,$ (27)

4.4. 粒子的几率波函数

5. 氢原子辐射实验对波函数的检验

5.1. 氢原子的吸收与辐射

5.2. 氢原子外层电子波函数方程

$V=abe\left(\frac{1}{{\lambda }_{0}}-\frac{1}{{\lambda }_{e}}\right),$ (28)

$V=\frac{{e}^{2}}{8\text{π}{\epsilon }_{0}{r}_{e}}\left(\sqrt{{\lambda }_{0}/{\lambda }_{e}}-\frac{1}{\sqrt{{\lambda }_{0}/{\lambda }_{e}}}\right),$ (29)

$\left[-\frac{{\beta }^{2}{h}_{m}^{2}}{2m+{m}_{m}}{\nabla }^{2}+V\left(r\right)\right]\psi =E\psi ,$ (30)

5.3. 氢原子外层电子几率波

6. 结论

Table 2. Data sheet of Hydrogen atom Palmer line system steady-state wave function and electron momentum probabilistic wave function

NOTES

*通讯作者。

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