Difference between revisions of "Variables Used in Elastic Scattering"

From New IAC Wiki
Jump to navigation Jump to search
(Blanked the page)
 
(34 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
=Lorentz Invariant Quantities=
 
−
==Total 4-Momentums==
 
−
As was [[DV_Calculations_of_4-momentum_components#4-Momentum_Invariants | shown earlier]] the scalar product of a 4-Momentum vector with itself ,
 
−
<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s</math></center> ,
 
  
−
and the length of a 4-Momentum vector composed of 4-Momentum vectors,
 
−
 
−
<center><math>{\mathbf P^2}=({\mathbf P_1}+{\mathbf P_2})^2=(E_1+E_2)^2-(\vec p_1 +\vec p_2 )^2=(m_1+m_2)^2=s</math></center>,
 
−
 
−
are invariant quantities.
 
−
 
−
It was [[DV_Calculations_of_4-momentum_components#Equal_masses | further shown ]] that
 
−
 
−
<center><math>{\mathbf P^*}^2={\mathbf P}^2</math></center>
 
−
 
−
 
−
<center>''where'' <math>{\mathbf P^*}=({\mathbf P_1^*}+{\mathbf P_2^*})^2</math> ''represents the 4-Momentum Vector in the CM frame''</center>
 
−
 
−
 
−
<center> ''and'' <math>{\mathbf P}=({\mathbf P_1}+{\mathbf P_2})^2</math> ''represents the 4-Momentum Vector in the initial Lab frame''</center>
 
−
 
−
which can be expanded to
 
−
 
−
<center><math>{\mathbf P^*}^2={\mathbf P}^2={\mathbf P^'}^2</math></center>
 
−
 
−
 
−
<center>''where'' <math>{\mathbf P^'}=({\mathbf P_1^'}+{\mathbf P_2^'})^2</math> ''represents the 4-Momentum Vector in the final Lab frame''</center>
 
−
 
−
==New 4-Momentum Quantities==
 
−
Working in just the Lab frame, we can form new 4-Momentum Vectors comprised of 4-Momenta in this frame, with
 
−
 
−
<center><math>{\mathbf P_1}- {\mathbf P_1^'}= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_a}</math></center>
 
−
 
−
 
−
<center><math>{\mathbf P_1}- {\mathbf P_2^'}= \left( \begin{matrix}E_1-E_2'\\ p_{1(x)}-p_{2(x)}^' \\ p_{1(y)}-p_{2(y)}^' \\ p_{1(z)}-p_{2(z)}^'\end{matrix} \right)={\mathbf P_b}</math></center>
 
−
 
−
 
−
<center><math>{\mathbf P_2}- {\mathbf P_1^'}= \left( \begin{matrix}E_2-E_1'\\ p_{2(x)}-p_{1(x)}^' \\ p_{2(y)}-p_{1(y)}^' \\ p_{2(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_c}</math></center>
 
−
 
−
 
−
<center><math>{\mathbf P_2}- {\mathbf P_2^'}= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)={\mathbf P_d}</math></center>
 
−
 
−
Using the algebraic fact
 
−
 
−
<center><math>\left({\mathbf a}- {\mathbf b}\right)^2=\left({\mathbf b}- {\mathbf a}\right)^2</math></center>
 
−
 
−
 
−
and the fact that the length of these 4-Momentum Vectors are invariant,
 
−
 
−
<center><math>\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left({\mathbf P_1}^2-2{\mathbf P_1}\cdot {\mathbf P_1^'}+ {\mathbf P_1^'}\right)= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_a}\right)^2=s</math></center>
 
−
 
−
 
−
<center><math>\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left({\mathbf P_1}^2-2{\mathbf P_1}\cdot {\mathbf P_2^'}+ {\mathbf P_2^'}\right)= \left( \begin{matrix}E_1-E_2'\\ p_{1(x)}-p_{2(x)}^' \\ p_{1(y)}-p_{2(y)}^' \\ p_{1(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_b}\right)^2=s</math></center>
 
−
 
−
 
−
<center><math>\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left({\mathbf P_2}^2-2{\mathbf P_2}\cdot {\mathbf P_1^'}+ {\mathbf P_1^'}\right)= \left( \begin{matrix}E_2-E_1'\\ p_{2(x)}-p_{1(x)}^' \\ p_{2(y)}-p_{1(y)}^' \\ p_{2(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_c}\right)^2=s</math></center>
 
−
 
−
 
−
<center><math>\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left({\mathbf P_2}^2-2{\mathbf P_2}\cdot {\mathbf P_2^'}+ {\mathbf P_2^'}\right)= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_d}\right)^2=s</math></center>
 
−
 
−
Using the fact that the scalar product of a 4-momenta with itself is invariant,
 
−
 
−
 
−
 
−
<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s</math></center>
 
−
 
−
 
−
We can simiplify the expressions
 
−
 
−
<center><math>\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_1^'}+  m_1^'\right)= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_a}\right)^2=s</math></center>
 
−
 
−
 
−
<center><math>\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_2^'}+  m_2^'\right)= \left( \begin{matrix}E_1-E_2'\\ p_{1(x)}-p_{2(x)}^' \\ p_{1(y)}-p_{2(y)}^' \\ p_{1(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_b}\right)^2=s</math></center>
 
−
 
−
 
−
<center><math>\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_1^'}+  m_1^'}\right)= \left( \begin{matrix}E_2-E_1'\\ p_{2(x)}-p_{1(x)}^' \\ p_{2(y)}-p_{1(y)}^' \\ p_{2(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_c}\right)^2=s</math></center>
 
−
 
−
 
−
<center><math>\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_2^'}+  m_2^'\right)= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_d}\right)^2=s</math></center>
 
−
 
−
=Mandelstam Representation=
 
−
 
−
[[File:Mandelstam.png | 400 px]]
 

Latest revision as of 19:10, 1 June 2017

Retrieved from "https://wiki.iac.isu.edu/index.php?title=Variables_Used_in_Elastic_Scattering&oldid=115545"