Length Contraction, Time Dilation, Mass, Momentum and Energy Equations, Particle and Antiparticle Potential Energy, Pair Production and Annihilation Energy Equations from Harmonic Oscillator Rest Energy Equation and New Relations from Uncertainty Principle

The harmonic oscillator total rest energy equation is derived from simple and quantum-mechanical harmonic oscillator equations. Length contraction, time dilation, relativistic mass, momentum, and energy equations for a particle, electron, and Planck particle are derived from harmonic oscillator total rest energy equation and new relations derived from the Heisenberg uncertainty principle. Particle and antiparticle, electron and positron potential energy equations; particle-antiparticle, electron-positron pair productions, and pair annihilations minimum energy equations are derived from harmonic oscillator total rest energy equation. The rest energy equation is derived from a nonrelativistic differential equation.

Introduction

The electron Compton radius or the reduced Compton wavelength, which is obtained from scattering experiments and is taken as the photon scattering radius of the electron is given by [1]–[3]:

(1) $$\begin{array}{r}{r}_C{m}_{e}c=\hslash \end{array}$$

where $r_C={\lambda }_C/2\pi$ is the electron Compton radius, ${\lambda }_C$ is the Compton wavelength, $m_e$ is the electron rest mass, $c$ is the speed of light in vacuum and $\hslash$ is the reduced Planck constant. From (1), (2) is written:

(2) $$\begin{array}{r}{m}_e{c}^2=h{f}_C\end{array}$$

where $h$ is the Planck constant and $f_C$ is the Compton frequency.

Van Belle [2], [4] proposed a two-dimensional harmonic oscillator model for zitterbewegung electron, and assumed that the free electron is a pointlike charge in an electromagnetic orbital oscillation of radius $a$ and rotates at tangential speed of $c=\omega a.$ He derived the total energy equation for each oscillator as:

(3) $$\begin{array}{r}\frac{1}{2}m{v}^2+\frac{1}{2}k{x}^2=\frac{1}{2}m{\omega }^2{a}^2=\frac{1}{2}m{c}^2\end{array}$$

where $\omega ={\left(k/m\right)}^{1/2}$ is the angular frequency, $m$ is the electron mass and $k$ is the force constant.

Daywitt [3] obtained (4) from the Planck mass $m_{\ast }=\sqrt{\hslash c/G}$ and the Planck length $r_{\ast }=\sqrt{\hslash G/c^3}$,

(4) $$\begin{array}{r}{r}_{\ast }{m}_{\ast }c=\hslash \end{array}$$

and derived the ground level Planck particle quantum harmonic oscillator total energy equation by using (4),

(5) $$\begin{array}{r}\frac{1}{2}{m}_{\ast }{v}_x^2+\frac{1}{2}{K}_{\ast }{x}^2=\frac{1}{2}{m}_{\ast }{\omega }_{\ast }^2{r}_{\ast }^2=\frac{1}{2}{m}_{\ast }{c}^2=\frac{1}{2}\hslash {\omega }_{\ast }\end{array}$$

where ${\omega }_{\ast }={\left(K_{\ast }/m_{\ast }\right)}^{1/2}=c/r_{\ast }=1/t_{\ast },K_{\ast }$ is the spring constant, $t_{\ast }=\sqrt{\hslash G/c^5}$ is the Planck time [5], [6].

In Section 2, Van Belle [4] and Daywitt [3] equations for electron and Planck particle harmonic oscillators are generalized and harmonic oscillator total rest energy equation is derived from simple and quantum-mechanical harmonic oscillators equations. Length contraction, time dilation, relativistic mass, momentum and energy equations for a particle, electron and Planck particle are derived from harmonic oscillator total rest energy equation and new relations derived from Heisenberg uncertainty principle. In Section 3, particle and antiparticle, electron and positron potential energy equations; particle-antiparticle, electron-positron pair productions and pair annihilations minimum energy equations are derived from harmonic oscillator total rest energy equation. In Section 4, rest energy equation is derived from nonrelativistic differential equation.

Derivations of Length Contraction, Time Dilation, Relativistic Mass, Momentum and Energy Equations from Harmonic Oscillator Total Rest Energy Equation and New Relations from Heisenberg Uncertainty Principle

Total mechanical energy in simple harmonic motion is given by [1, p. 446]:

(6) $$\begin{array}{r}\frac{1}{2}m{v}_x^2+\frac{1}{2}{k}^{,}{x}^2=\frac{1}{2}{k}^{,}{A}^2=\text{Constant}\end{array}$$

where $x$ is the displacement from equilibrium, $A$ is the maximum displacement or the amplitude of the motion, $k^{,}$ is the force constant and $m$ is the particle mass. Since the motion is one-dimentional, $v^2=v_x^2$. When the particle reaches the point $x=A$, it comes momentarily to rest before reversing direction. That is, when $x=A$ (or $-A)$, $v_x=0$. At this rest point the energy is entirely potential and constant $1/2k^{,}{A}^2$, and is equal to the rest energy $1/2m{c}^2$. From these expressions, (7) can be written as:

(7) $$\begin{array}{r}\frac{1}{2}{k}^{\prime }{A}^2=\frac{1}{2}m{c}^2\end{array}$$

The relationship between the total energies of the simple and the ground level quantum-mechanical harmonic oscillators is given by [1, p. 1353]:

(8) $$\begin{array}{r}\frac{1}{2}{k}^{,}{A}^2=\frac{1}{2}\hslash \omega \end{array}$$

where $\omega ={\left(k^{\prime }/m\right)}^{1/2}=2\pi f$ and $f$ is frequency. Equation (8) shows that the energy of the photon is not equal to the energy of the ground level, but rather it is the energy difference between two levels. From (6)–(8), harmonic oscillator total rest energy eqution is obtained.

(9) $$\begin{array}{r}\frac{1}{2}m{v}_x^2+\frac{1}{2}{k}^{,}{x}^2=\frac{1}{2}{k}^{,}{A}^2=\frac{1}{2}\hslash \omega =\frac{1}{2}m{c}^2\end{array}$$

When the particle is at its maximum displacement ($x=\pm A)$ and instantaneously at rest, $v_x=0.$ When the particle is at equilibrium ($x=0$) and moving at its maximum speed, $v_x=v_{max}=c,$ and the maximum momentum of the particle is $p_{max}=m{v}_{max}=mc$, where $m$ is the rest mass [1, p.1354].

Heisenberg uncertainty principal for position and momentum is defined as [1, p.1354].

(10) $$\begin{array}{r}\mathrm{\Delta }x\mathrm{\Delta }{p}_x=\frac{\hslash }{2}\end{array}$$

The uncertainties in the particle’s position and momentum (calculated as standart deviations) for harmonic oscillator are given $\mathrm{\Delta }x=A/\sqrt{2}$ and $\mathrm{\Delta }{p}_x=p_{max}/\sqrt{2}$, respectively, [1, p.1354]. When these uncertainties are substituted in (10), (11) is obtained.

(11) $$\begin{array}{r}Amc=\hslash \end{array}$$

From (9), (12) is obtained.

(12) $$\begin{array}{r}x=A\sqrt{1-\frac{v_x^2}{c^2}}\end{array}$$

where $A=c\sqrt{m/k^{\prime }}$. The positive square root of $x$ is taken since $x$ must be positive. When $x=l$, $A=l_0$ and $v_x=v$ are substituted in (12), length contraction is obtained [1, p. 1234]:

(13) $$\begin{array}{r}l=l_0\sqrt{1-\frac{v^2}{c^2}}\end{array}$$

When (12) is substituted in (11), (14) is obtained:

(14) $$\begin{array}{r}Amc=x\frac{m}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}c=x{m}_{rel}c=\hslash \end{array}$$

where $v_x=v$ and $m_{rel}$ is the relativistic mass [1, p. 1244]:

(15) $$\begin{array}{r}{m}_{rel}=\frac{m}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When (15) is multiplied by $v$, relativistic momentum $p$ is obtained [1, p. 1243]:

(16) $$\begin{array}{r}p=m_{rel}v=\frac{mv}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When (14) is multiplied by $c$, (17) is obtained:

(17) $$\begin{array}{r}Am{c}^2=x\frac{m{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}=xE=\hslash c\end{array}$$

where $E$ is the total energy [1, p. 1247]:

(18) $$\begin{array}{r}E=\frac{m{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When $v=0$ is substituted in (18), rest energy $E_0$ is obtained:

(19) $$\begin{array}{r}{E}_0=m{c}^2\end{array}$$

From (18) and (19), relativistic kinetic energy $K$ is obtained:

(20) $$\begin{array}{r}K=E-E_0=\frac{m{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}-m{c}^2\end{array}$$

From (1), (4) and (11), the relation can be written as follows:

(21) $$\begin{array}{r}{r}_C{m}_{e}c=r_{\ast }{m}_{\ast }c=Amc=\hslash \end{array}$$

When $m=m_e$ is substituted in (21), $A=r_C$ is obtained. When $A=r_C$ and $x=x_C$ are substituted in (12), contraction of Compton radius [7] is obtained:

(22) $$\begin{array}{r}{x}_C=r_C\sqrt{1-\frac{v^2}{c^2}}\end{array}$$

When (22) is substituted in (1), the relation is obtained for electron:

(23) $$\begin{array}{r}{r}_C{m}_{e}c=x_C\frac{m_e}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}c=x_C{m}_{e_{rel}}c=\hslash \end{array}$$

where $m_{e_{rel}}$ is the relativistic mass of electron:

(24) $$\begin{array}{r}{m}_{e_{rel}}=\frac{m_e}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When (24) is multiplied by $v$, relativistic momentum $p_e$ of electron is obtained:

(25) $$\begin{array}{r}{p}_e=m_{e_{rel}}v=\frac{m_{e}v}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When (23) is multiplied by $c$, (26) is obtained:

(26) $$\begin{array}{r}{r}_C{m}_e{c}^2=x_C\frac{m_e{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}=x_C{E}_e=\hslash c\end{array}$$

where $E_e$ is the total energy of electron.

(27) $$\begin{array}{r}{E}_e=\frac{m_e{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When $v=0$ is substituted in (27), rest energy $E_{e_0}$ of electron is obtained:

(28) $$\begin{array}{r}{E}_{e_0}=m_e{c}^2\end{array}$$

Relativistic kinetic energy of electron $K_e$ is obtained from (27) and (28):

(29) $$\begin{array}{r}{K}_e=\frac{m_e{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}-m_e{c}^2\end{array}$$

When $m=m_{\ast }$ is substituted in (21), $A=r_{\ast }$ is obtained. When $A=r_{\ast }$ and $x=x_{\ast }$ are substituted in (12), contraction of Planck length is obtained:

(30) $$\begin{array}{r}{x}_{\ast }=r_{\ast }\sqrt{1-\frac{v^2}{c^2}}\end{array}$$

When (30) is substituted in (4), the relation is obtained for Planck particle:

(31) $$\begin{array}{r}{r}_{\ast }{m}_{\ast }c=x_{\ast }\frac{m_{\ast }}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}c=x_{\ast }{m}_{{\ast }_{rel}}c=\hslash \end{array}$$

where $m_{{\ast }_{rel}}$ is the relativistic mass of Planck particle.

(32) $$\begin{array}{r}{m}_{{\ast }_{rel}}=\frac{m_{\ast }}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When (32) is multiplied by $v$, relativistic momentum $p_{\ast }$ of Planck particle is obtained:

(33) $$\begin{array}{r}{p}_{\ast }=m_{{\ast }_{rel}}v=\frac{m_{\ast }v}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When (31) is multiplied by $c$, (34) is obtained:

(34) $$\begin{array}{r}{r}_{\ast }{m}_{\ast }{c}^2=x_{\ast }\frac{m_{\ast }{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}=x_{\ast }{E}_{\ast }=\hslash c\end{array}$$

where $E_{\ast }$ is the total energy of Planck particle:

(35) $$\begin{array}{r}{E}_{\ast }=\frac{m_{\ast }{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When $v=0$ is substituted in (35), rest energy $E_{{\ast }_0}$ of Planck particle is obtained.

(36) $$\begin{array}{r}{E}_{{\ast }_0}=m_{\ast }{c}^2\end{array}$$

Relativistic kinetic energy of Planck particle $K_{\ast }$ is obtained from (35) and (36):

(37) $$\begin{array}{r}{K}_{\ast }=\frac{m_{\ast }{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}-m_{\ast }{c}^2\end{array}$$

From $r_{\ast }$ and $t_{\ast }$, (38) is obtained:

(38) $$\begin{array}{r}{r}_{\ast }{t}_{\ast }=\frac{G}{c^4}\hslash \end{array}$$

From (21) and (38)–(40) are obtained:

(39) $$\begin{array}{r}{r}_{\ast }{t}_{\ast }=r_{\ast }\frac{m_{\ast }G}{c^3}=r_C\frac{m_{e}G}{c^3}=A\frac{mG}{c^3}=\frac{G}{c^4}\hslash \end{array}$$

(40) $$\begin{array}{r}{r}_{\ast }{t}_{\ast }=r_C{t}_C=A{t}_0=\frac{G}{c^4}\hslash \end{array}$$

where $t_{\ast }=m_{\ast }G/c^3$, $t_C=m_{e}G/c^3$ and $t_0=mG/c^3$ are the Planck, electron and particle times. When (12) is substituted in (40), (41) is obtained:

(41) $$\begin{array}{r}A{t}_0=x\frac{t_0}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}=xt=\frac{G}{c^4}\hslash \end{array}$$

where $t$ is time dilation for particle [1, p. 1229], [8, p. 209]:

(42) $$\begin{array}{r}t=\frac{t_0}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When (22) is substituted in (40), (43) is obtained:

(43) $$\begin{array}{r}{x}_C\frac{t_C}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}=x_C{t}_{C_{rel}}=\frac{G}{c^4}\hslash \end{array}$$

where $t_{C_{rel}}$ is time dilation for electron:

(44) $$\begin{array}{r}{t}_{C_{rel}}=\frac{t_C}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

When (30) is substituted in (40), (45) is obtained:

(45) $$\begin{array}{r}{x}_{\ast }\frac{t_{\ast }}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}=x_{\ast }{t}_{{\ast }_{rel}}=\frac{G}{c^4}\hslash \end{array}$$

where $t_{{\ast }_{rel}}$ is time dilation for Planck particle:

(46) $$\begin{array}{r}{t}_{{\ast }_{rel}}=\frac{t_{\ast }}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\end{array}$$

Derivations of Particle and Antiparticle, Electron and Positron Potential Energy Equations, Particle-Antiparticle, Electron-Positron Pair Productions and Pair Annihilations Minimum Energy Equations from Harmonic Oscillator Total Rest Energy Equation

Particle and antiparticle can be treated as harmonic oscillators. Potential energy equation for a particle can be written from (9) as:

(47) $$\begin{array}{r}2U=k^{,}{x}^2=m{c}^2-m{v}^2\end{array}$$

where $x=0$, $2U=0$, and $v=c$. When $x$ increases, $v$ decreases [1, p.1353] and $2U$ increases. When $x=A$, $v=0$, $2U=2U_{max}$, and (48) can be written from (47).

(48) $$\begin{array}{r}2U_{max}=k^{,}{A}^2=m{c}^2\end{array}$$

From (47), potential energy equation for electron can be written as:

(49) $$\begin{array}{r}2U_{e^{-}}=k_e^{\prime }{x}^2=m_e{c}^2-m_e{v}^2\end{array}$$

When $x=0$, $2U_{e^{-}}=0$, and $v=c$. When $x$ increases, $v$ decreases and 2$U_{e^{-}}$ increases. When $x=A=r_C$, $v=0$, $2U_{e^{-}}=2U_{e_{max}^{-}}$, and (50) can be written from (49):

(50) $$\begin{array}{r}2U_{e_{max}^{-}}=k_e^{\prime }{r}_C^2=m_e{c}^2\end{array}$$

According to the symmetry property of (47), potential energy equation for an antiparticle can be written as:

(51) $$\begin{array}{r}-2U=-k^{,}{x}^2=-m{c}^2+m{v}^2\end{array}$$

When $x=0$, $-2U=0$, and $v=c.$ When $x$ increases, $v$ decreases and $-2U$ increases. When $x=A$, $v=0,$ $-2U$ = $-2U_{max}$, and (52) can be written from (51):

(52) $$\begin{array}{r}-2U_{max}=-k^{,}{A}^2=-m{c}^2\end{array}$$

The minus sign shows the opposite direction of antiparticle to particle. From (51), potential energy equation for positron can be written as:

(53) $$\begin{array}{r}2U_{e^{+}}=-2U_{e^{-}}=-k_e^{\prime }{x}^2=-m_e{c}^2+m_e{v}^2\end{array}$$

When $x=0$, $-2U_{e^{-}}=0$, and $v=c$. When $x$ increases, $v$ decreases and $-2U_{e^{-}}$ increases. When $x=A=r_C$, $v=0$, $-2U_{e^{-}}=-2U_{e_{max}^{-}}$, and (54) can be written from (53):

(54) $$\begin{array}{r}2U_{e_{max}^{+}}=-2U_{e_{max}^{-}}=-k_e^{\prime }{r}_C^2=-m_e{c}^2\end{array}$$

Equations (53) and (54) show that positron is a real particle and not a hole or vacancy [1, p.1482], [9], and it is not an electron traveling backward in time [10].

The minimum energy for particle-antiparticle pair production is obtained from the difference between (48) and (52) as:

(55) $$\begin{array}{r}4U_{max}=2k^{\prime }{A}^2=2m{c}^2\end{array}$$

The minimum energy for electron-positron pair production is obtained from the difference between (50) and (54) as:

(56) $$\begin{array}{r}2U_{e_{max}^{-}}-2U_{e_{max}^{+}}=4U_{e_{max}^{-}}=2k_e^{\prime }{r}_C^2=2m_e{c}^2\end{array}$$

Fig. 1 shows the potential energies for electron $e_{-}$ and positron $e_{+}$ as functions of position $x$. The curves for electron and positron are symmetric about the $x$-axis and opposite directions due to the energy and momentum conservations.

Fig. 1. Potential energies $2U_e$ and $-2U_e$ for electron $e^{-}$ and positron $e^{+}$ as functions of position $x.$

From (8), (48) and (52), (57) and (58) can be written as:

(57) $$\begin{array}{r}2U_{max}=m{c}^2=hf=pc\end{array}$$

(58) $$\begin{array}{r}-2U_{max}=-m{c}^2=-hf=-pc\end{array}$$

where $p$ is the photon momentum. The energy produced by pair annihilation is obtained from the difference between (57) and (58) as:

(59) $$\begin{array}{r}4U_{max}=2m{c}^2=2hf=2pc\end{array}$$

From (57) and (58), (60) and (61) can be written for electron and positron as:

(60) $$\begin{array}{r}2U_{e^{-}max}=m_e{c}^2=h{f}_C=p_{C}c\end{array}$$

(61) $$\begin{array}{r}2U_{e_{max}^{+}}=-2U_{e^{-}max}=-m_e{c}^2=-h{f}_C=-p_{C}c\end{array}$$

where $p_C$ is the Compton momentum. The energy produced by the electron-positron pair annihilation is obtained from the difference between (60) and (61) as:

(62) $$\begin{array}{r}2U_{e_{max}^{-}}-2U_{e_{max}^{+}}=4U_{e_{max}^{-}}=2m_e{c}^2=2h{f}_C=2p_{C}c\end{array}$$

Two photons have equal and opposite momenta by momentum conservation and have equal energies.

Derivation of Rest Energy from Differential Equation

Einstein derivations of rest energy are based on the radiation pressure [8, p.232–234] and special relativity [11, p.49,50]. In this study, rest energy is derived from differential equation.

For nonrelativistic speed ($v\ll c$), the rate of change of total energy $E$ with velocity $v$ can be given as:

(63) $$\begin{array}{r}\frac{dE}{dv}=aEv\end{array}$$

From (63), (64) can be written:

(64) $$\begin{array}{r}{\int }_{E_0}^E\frac{dE}{E}=a{\int }_0^{v}vdv\end{array}$$

where $E=E_0+K$ is the total energy, $E_0$ is the rest energy, $K=m{v}^2/2$ is the kinetic energy $m$ is the rest mass and $a$ is a constant. If $a=1/c^2$ is substituted in (64), (65) is obtained:

(65) $$\begin{array}{r}\frac{E}{E_0}=e^{\frac{v^2}{2c^2}}\end{array}$$

From (65), (66) can be written:

(66) $$\begin{array}{r}\frac{E_0+K}{E_0}=1+\frac{v^2}{2c^2}+\dots \end{array}$$

When $K=m{v}^2/2$ is substituted in (66), the rest energy is obtained:

(67) $$\begin{array}{r}{E}_o=m{c}^2\end{array}$$

Conclusion

The harmonic oscillator total rest energy equation was derived from simple and quantum-mechanical harmonic oscillator equations. New relations were derived from Heisenberg’s uncertainty principle. Length contraction, time dilation, relativistic mass, momentum, and energy equations for a particle, electron, and Planck particle were derived from harmonic oscillator total rest energy equation and new relations. Particle and antiparticle, electron and positron potential energy equations; particle-antiparticle, electron-positron pair productions, and pair annihilations energy equations were derived from harmonic oscillator total rest energy equation. The rest energy equation was derived from a nonrelativistic differential equation.