New 4-momentum quantities - Revision history

Vanwdani: /* New 4-Momentum Quantities */

2017-06-08T20:07:41Z

New 4-Momentum Quantities

Vanwdani: /* New 4-Momentum Quantities */

2017-06-02T09:48:01Z

New 4-Momentum Quantities

Vanwdani: Created page with "=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 ${\mathbf P_1^*}- {\mathbf P…"$

2017-05-31T20:42:55Z

Created page with "=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…"

New page

=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>

← Older revision		Revision as of 20:07, 8 June 2017
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−	~~=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>~~

@@ Line 20: / Line 20: @@
 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_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^{'*}}\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_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^{'*}}\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_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^{'*}}\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>
+<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,

New 4-momentum quantities - Revision history

Vanwdani: /* New 4-Momentum Quantities */

Vanwdani: /* New 4-Momentum Quantities */

Vanwdani: Created page with "=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 {\mathbf P_1^*}- {\mathbf P…"

Vanwdani: Created page with "=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 ${\mathbf P_1^*}- {\mathbf P…"$