Difference between revisions of "Relativistic Units"

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<center><math>\underline{\textbf{Navigation}}</math>
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[[Relativistic_Frames_of_Reference|<math>\vartriangleleft </math>]]
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[[VanWasshenova_Thesis#Moller_Scattering|<math>\triangle </math>]]
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[[4-vectors|<math>\vartriangleright </math>]]
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</center>
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=Relativistic Units=
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From the definition of 4-vectors shown earlier, we know that  
 
From the definition of 4-vectors shown earlier, we know that  
  
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The 4-vectors and 4-momenta are defined to be in units of distance and momentum and as such must be multiplied or divided respectively by the speed of light to meet this requirement.  For simplicity, the units of c can be chosen to be 1.
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The 4-vectors and 4-momenta are defined to be in units of distance and momentum and as such must be multiplied or divided respectively by the speed of light to meet this requirement.  For simplicity, the units of c can be chosen to be 1. This implies:
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<center><math>c=1=\frac{length}{time}</math></center>
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<center><math>\therefore\ length\ units=time\ units</math></center>
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This also implies that from the definition of an electromagnetic wave
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<center><math>c \equiv \frac{1}{\sqrt{\epsilon_0 \mu_0}} \rightarrow \epsilon_0=\mu_0=1</math></center>
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The relativistic equation for energy
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<center><math>E^2 \equiv m^2c^4+p^2c^2</math></center>
  
  
DeBroglie's equation and the wave number can be used to rewrite the 4-momenta vectors
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<center><math>\rightarrow E^2 = m^2+p^2</math></center>
  
<center><math>E=\hbar \omega \qquad k=\frac{p}{\hbar} \rightarrow p=k \hbar</math></center>
 
  
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<center><math>\therefore\ energy\ units=mass\ units=momentum\ units</math></center>
  
  
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The Planck-Einstein relation and the de Broglie relation can be used to substitute into the relativistic energy equation
  
  
<center><math>
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<center><math>E^2 = m^2+p^2</math></center>
\mathbf{P} \equiv
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\begin{bmatrix}
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p^0 \\
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p^1 \\
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<center><math>\rightarrow E=\hbar \omega \qquad \qquad p=k \hbar</math></center>
p^2 \\
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p^3
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\end{bmatrix}=
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<center><math> \hbar^2 \omega^2 = m^2+k^2 \hbar^2</math></center>
\begin{bmatrix}
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\frac{E}{c} \\
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p_x \\
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p_y \\
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Since the units of <math>\omega =\frac{1}{time}\ </math> and the units of <math>k=\frac{1}{length}</math> setting <math>\hbar=1</math> will preserve the relationship
p_z
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\end{bmatrix}=
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\begin{bmatrix}
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\frac{\hbar \omega}{c} \\
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<center><math> length\ units=time\ units</math></center>
\hbar k_x \\
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\hbar k_y \\
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\hbar k_z
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<center><math>\rightarrow Energy\ units=mass\ units=momentum\ units=\frac{1}{length\ units}=\frac{1}{time\ units}</math></center>
\end{bmatrix}\rightarrow
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\mathbf{\frac{P}{\hbar}}\equiv
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\mathbf{K} \equiv  
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\begin{bmatrix}
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k^0 \\
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The amount of energy gained by a charged particle moving across an electric potential of 1 volt are declared to be electron-volts
k^1 \\
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k^2 \\
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k^3
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<center><math>1eV \equiv (1V)(1e^-)= \frac{1J}{1C}(1.6021766208(98)\times 10^{-19} C)=1.6021766208(98)\times 10^{-19} J</math></center>
\end{bmatrix}=
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\begin{bmatrix}
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\frac{\omega}{c} \\
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<center><math>\hbar \equiv 1.054571800(13)\times 10^{-34} J \cdot s \qquad c \equiv 2.99792458\frac{m}{s}</math></center>
k_x \\
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k_y \\
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k_z
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<center><math>\hbar c = 3.16152649\times 10^{-26} J\cdot m</math></center>
\end{bmatrix}</math></center>
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Converting to eV
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<center><math>\frac{3.16152649\times 10^{-26} J\cdot m}{1.6021766208(98)\times 10^{-19} J}=\frac{1.9732696\times 10^{-7} eV \cdot m</math></center>
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----
  
<math>\hbar</math> in SI units is defined as:
 
  
<center><math>\hbar \approx 1.0545718 \times 10^{-34} m \cdot kg \cdot \frac{m}{s}</math></center>
 
  
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<center><math>\underline{\textbf{Navigation}}</math>
  
Since c is already to be defined as equal to zero, this implies unit of mass must also be equal to one.  By convention, the mass of the proton
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[[Relativistic_Frames_of_Reference|<math>\vartriangleleft </math>]]
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[[VanWasshenova_Thesis#Moller_Scattering|<math>\triangle </math>]]
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[[4-vectors|<math>\vartriangleright </math>]]
  
<center><math>M_{p} \equiv 1.6726219 \times 10^{-27} kg =1</math></center>
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</center>

Latest revision as of 18:46, 15 May 2018

[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]

Relativistic Units

From the definition of 4-vectors shown earlier, we know that

[math]\mathbf{R} \equiv \begin{bmatrix} x^0 \\ x^1 \\ x^2 \\ x^3 \end{bmatrix}= \begin{bmatrix} ct \\ x \\ y \\ z \end{bmatrix} \qquad \qquad \mathbf{P} \equiv \begin{bmatrix} p^0 \\ p^1 \\ p^2 \\ p^3 \end{bmatrix}= \begin{bmatrix} \frac{E}{c} \\ p_x \\ p_y \\ p_z \end{bmatrix}[/math]


The 4-vectors and 4-momenta are defined to be in units of distance and momentum and as such must be multiplied or divided respectively by the speed of light to meet this requirement. For simplicity, the units of c can be chosen to be 1. This implies:

[math]c=1=\frac{length}{time}[/math]


[math]\therefore\ length\ units=time\ units[/math]


This also implies that from the definition of an electromagnetic wave


[math]c \equiv \frac{1}{\sqrt{\epsilon_0 \mu_0}} \rightarrow \epsilon_0=\mu_0=1[/math]


The relativistic equation for energy

[math]E^2 \equiv m^2c^4+p^2c^2[/math]


[math]\rightarrow E^2 = m^2+p^2[/math]


[math]\therefore\ energy\ units=mass\ units=momentum\ units[/math]


The Planck-Einstein relation and the de Broglie relation can be used to substitute into the relativistic energy equation


[math]E^2 = m^2+p^2[/math]


[math]\rightarrow E=\hbar \omega \qquad \qquad p=k \hbar[/math]


[math] \hbar^2 \omega^2 = m^2+k^2 \hbar^2[/math]


Since the units of [math]\omega =\frac{1}{time}\ [/math] and the units of [math]k=\frac{1}{length}[/math] setting [math]\hbar=1[/math] will preserve the relationship


[math] length\ units=time\ units[/math]


[math]\rightarrow Energy\ units=mass\ units=momentum\ units=\frac{1}{length\ units}=\frac{1}{time\ units}[/math]



The amount of energy gained by a charged particle moving across an electric potential of 1 volt are declared to be electron-volts


[math]1eV \equiv (1V)(1e^-)= \frac{1J}{1C}(1.6021766208(98)\times 10^{-19} C)=1.6021766208(98)\times 10^{-19} J[/math]


[math]\hbar \equiv 1.054571800(13)\times 10^{-34} J \cdot s \qquad c \equiv 2.99792458\frac{m}{s}[/math]


[math]\hbar c = 3.16152649\times 10^{-26} J\cdot m[/math]


Converting to eV

[math]\frac{3.16152649\times 10^{-26} J\cdot m}{1.6021766208(98)\times 10^{-19} J}=\frac{1.9732696\times 10^{-7} eV \cdot m[/math]




[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]