Difference between revisions of "Variables Used in Elastic Scattering"

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-=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={\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>
-<center>''and'' <math>{\mathbf P^{'*}}=({\mathbf P_1^{'*}}+{\mathbf P_2^{'*}})^2</math> ''represents the 4-Momentum Vector in the final CM frame''</center>
-==New 4-Momentum Quantities==
-Working in just the CM 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</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</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</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</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</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^{'*2}\right)=\left({\mathbf P_a^*}\right)^2</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^{'*2}\right)=\left({\mathbf P_b^*}\right)^2</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^{'*2}\right)=\left({\mathbf P_c^*}\right)^2</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^{'*2}\right)=\left({\mathbf P_d^*}\right)^2</math></center>
-Finding the cross terms,
-<center><math>{\mathbf P_1^*}\cdot {\mathbf P^{'*}}=\left(\begin{matrix} E_1^*\\ p_{1(x)}^* \\ p_{1(y)}^* \\ p_{1(z)}^* \end{matrix} \right)\cdot \left( \begin{matrix}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 &0 & 0 &-1\end{matrix} \right)\cdot \left(\begin{matrix} E^{'*} & p_{1(x)}^{'*} & p_{1(y)}^{'*} & p_{1(z)}^{'*} \end{matrix} \right)=E_1^*E_1^{'*}-\vec p_1^*\cdot \vec p_1^{'*} </math></center>
-<center><math>\left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_1^{'*}-2\vec p_1^*\cdot \vec p_1^{'*}}+  m_1^{'*2}\right)=\left({\mathbf P_a^*}\right)^2</math></center>
-<center><math>\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_2^{'*}-2\vec p_1^*\cdot \vec p_2^{'*}}+  m_2^{'*2}\right)=\left({\mathbf P_b^*}\right)^2</math></center>
-<center><math>\left({\mathbf P_2^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_1^{'*}-2\vec p_2^*\cdot \vec p_1^{'*}}+  m_1^{'*2}\right)=\left({\mathbf P_c^*}\right)^2</math></center>
-<center><math>\left({\mathbf P_2^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_2^{'*}-2\vec p_2^*\cdot \vec p_2^{'*}}+  m_2^{'*2}\right)=\left({\mathbf P_d^*}\right)^2</math></center>
-Using the fact that in the CM frame,
-<center>[[File:CM.png |400 px]]</center>
-<center><math>\vec p_1^*=-\vec p_2^*</math></center>
-<center><math>\vec p_1^{'*}=-\vec p_2^{'*}</math></center>
-Since this is an ellastic collision between identical particles, Energy is conserved,
-<center><math>E_1^*=E_1^{'*}</math></center>
-<center><math>E_2^*=E_2^{'*}</math></center>
-Lastly as [[DV_Calculations_of_4-momentum_components#Scattered_and_Moller_Electron_energies_in_CM |shown earlier]], <math>E_1^*=E_2^*</math>
-We can further simplify
-<center><math>\left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_1^{*2}-2{E_2^*E_2^{'*}-2\vec p_2^*\cdot \vec p_2^{'*}}+  m_1^{'*2}\right)=\left({\mathbf P_a^*}\right)^2</math></center>
-<center><math>\left({\mathbf P_2^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_2^{'*}-2\vec p_2^*\cdot \vec p_2^{'*}}+  m_2^{'*2}\right)=\left({\mathbf P_d^*}\right)^2</math></center>
-<center><math>\Longrightarrow \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^*}- {\mathbf P_2^{'*}}\right)^2</math></center>
-<center><math>\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_1^{*2}-2{E_2^*E_1^{'*}-2\vec p_2^*\cdot \vec p_1^{'*}}+  m_2^{'*2}\right)=\left({\mathbf P_b^*}\right)^2</math></center>
-<center><math>\left({\mathbf P_2^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_1^{'*}-2\vec p_2^*\cdot \vec p_1^{'*}}+  m_1^{'*2}\right)=\left({\mathbf P_c^*}\right)^2</math></center>
-<center><math>\Longrightarrow \left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left({\mathbf P_2^*}- {\mathbf P_1^{'*}}\right)^2</math></center>
-=Mandelstam Representation=
-<center>[[File:Mandelstam.png | 400 px]]</center>
-<center><math>\Longrightarrow \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2\equiv s</math></center>
-<center><math>\Longrightarrow \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^*}- {\mathbf P_2^{'*}}\right)^2\equiv  t</math></center>
-<center><math>\Longrightarrow \left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left({\mathbf P_2^*}- {\mathbf P_1^{'*}}\right)^2\equiv s</math></center>

Difference between revisions of "Variables Used in Elastic Scattering"

Latest revision as of 19:10, 1 June 2017

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