Difference between revisions of "Relativistic Units"

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Line 33: Line 33:
  
  
DeBroglie's equation
+
DeBroglie's equation and the wave number can be used to rewrite the 4-momenta vectors
  
<center><math>E=\hbar \omega \rightarrow \mathbf P=\hbar \mathbf K</math></center>
+
<center><math>E=\hbar \omega \qquad k=\frac{p}{\hbar} \rightarrow p=k \hbar</math></center>
  
  
  
  
<center><math>\mathbf{K} \equiv  
+
 
 +
<center><math>
 +
\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}=
 +
\begin{bmatrix}
 +
\frac{\hbar \omega}{c} \\
 +
\hbar k_x \\
 +
\hbar k_y \\
 +
\hbar k_z
 +
\end{bmatrix}
 +
\mathbf{K} \equiv  
 
\begin{bmatrix}
 
\begin{bmatrix}
 
k^0 \\
 
k^0 \\

Revision as of 15:59, 27 June 2017

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.


DeBroglie's equation and the wave number can be used to rewrite the 4-momenta vectors

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



[math] \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}= \begin{bmatrix} \frac{\hbar \omega}{c} \\ \hbar k_x \\ \hbar k_y \\ \hbar k_z \end{bmatrix} \mathbf{K} \equiv \begin{bmatrix} k^0 \\ k^1 \\ k^2 \\ k^3 \end{bmatrix}= \begin{bmatrix} \frac{\omega}{c} \\ k_x \\ k_y \\ k_z \end{bmatrix}[/math]