View source for New 4-momentum quantities

=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^{'*}}^2 \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^{'*}}^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^{'*}}^2 \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^{'*}}^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>