Special Case of Equal Mass Particles - Revision history

Vanwdani at 18:53, 15 May 2018

2018-05-15T18:53:50Z

Vanwdani at 01:03, 16 June 2017

2017-06-16T01:03:41Z

Vanwdani at 22:32, 15 June 2017

2017-06-15T22:32:52Z

Vanwdani at 22:29, 15 June 2017

2017-06-15T22:29:25Z

Vanwdani at 22:11, 15 June 2017

2017-06-15T22:11:14Z

Vanwdani: /* Special Case of Equal Mass Particles */

2017-06-14T02:15:11Z

Special Case of Equal Mass Particles

Vanwdani: Created page with "=Special Case of Equal Mass Particles= For incoming electrons moving only in the z-direction, we can write ${\mathbf P_1}+ {\mathbf P_2}= \left( \begin{matrix}E_1…"$

2017-05-31T17:55:46Z

Created page with "=Special Case of Equal Mass Particles= For incoming electrons moving only in the z-direction, we can write <center><math>{\mathbf P_1}+ {\mathbf P_2}= \left( \begin{matrix}E_1…"

New page

=Special Case of Equal Mass Particles=
For incoming electrons moving only in the z-direction, we can write

<center><math>{\mathbf P_1}+ {\mathbf P_2}= \left( \begin{matrix}E_1+E_2\\ 0 \\ 0 \\ p_{1(z)}+p_{2(z)}\end{matrix} \right)={\mathbf P}</math></center>

We can perform a Lorentz transformation to the Center of Mass frame, with zero total momentum

<center><math>\left( \begin{matrix}E^*_{1}+E^*_{2}\\ 0 \\ 0 \\ 0\end{matrix} \right)=\left(\begin{matrix}\gamma^* & 0 & 0 & -\beta^* \gamma^*\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ -\beta^* \gamma^* & 0 & 0 & \gamma^* \end{matrix} \right) . \left( \begin{matrix}E_{1}+E_{2}\\ 0 \\ 0 \\ p_{1(z)}+p_{2(z)}\end{matrix} \right)</math></center>

Without knowing the values for gamma or beta, we can utalize the fact that lengths of the two 4-momenta are invariant

<center><math>s={\mathbf P^*}^2=(E^*_{1}+E^*_{2})^2-(\vec p\ ^*_{1}+\vec p\ ^*_{2})^2=(m_{1}^*+m_{2}^*)^2</math></center>

<center><math>s={\mathbf P}^2=(E_{1}+E_{2})^2-(\vec p_{1}+\vec p_{2})^2=(m_{1}+m_{2})^2</math></center>

This gives,
<center><math>\Longrightarrow (m_{1}^*+m_{2}^*)^2=(m_{1}+m_{2})^2</math></center>

Using the fact that

<center><math>\begin{cases}
m_{1}=m_{2} \\
m_{1}^*=m_{2}^*
\end{cases}</math></center>
since the rest mass energy of the electrons remains the same in inertial frames.

Substituting, we find
<center><math>(m_{1}^*+m_{1}^*)^2=(m_{1}+m_{1})^2</math></center>

<center><math>2m_{1}^*=2m_{1}</math></center>

<center><math>m_{1}^*=m_{1}</math></center>

<center><math>\Longrightarrow m_{1}=m^*_{1}\ ; m_{2}=m^*_{2}</math></center>

This confirms that the mass remains constant between the frames of reference.

@@ Line 1: / Line 1: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Initial_CM_Frame_4-momentum_components|<math>\vartriangleleft </math>]]
@@ Line 71: / Line 71: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Initial_CM_Frame_4-momentum_components|<math>\vartriangleleft </math>]]

← Older revision		Revision as of 01:03, 16 June 2017
Line 63:		Line 63:

	This confirms that the mass remains constant between the frames of reference.		This confirms that the mass remains constant between the frames of reference.
		+
		+
		+
		+
		+	----
		+
		+
		+
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[Initial_CM_Frame_4-momentum_components\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Initial_4-momentum_Components\|<math>\triangle </math>]]
		+	[[Total_Energy_in_CM_Frame\|<math>\vartriangleright </math>]]
		+
		+	</center>

@@ Line 1: / Line 1: @@
 <center><math>\textbf{\underline{Navigation}}</math>
-[[Limits_based_on_Mandelstam_Variables|<math>\vartriangleleft </math>]]
+[[Initial_CM_Frame_4-momentum_components|<math>\vartriangleleft </math>]]
 [[VanWasshenova_Thesis#Initial_4-momentum_Components|<math>\triangle </math>]]
-[[Special_Case_of_Equal_Mass_Particles|<math>\vartriangleright </math>]]
+[[Total_Energy_in_CM_Frame|<math>\vartriangleright </math>]]
 </center>

@@ Line 2: / Line 2: @@
 [[Limits_based_on_Mandelstam_Variables|<math>\vartriangleleft </math>]]
-[[VanWasshenova_Thesis#Initial_Lab_Frame_4-momentum_components|<math>\triangle </math>]]
+[[VanWasshenova_Thesis#Initial_4-momentum_Components|<math>\triangle </math>]]
 [[Special_Case_of_Equal_Mass_Particles|<math>\vartriangleright </math>]]

← Older revision		Revision as of 22:11, 15 June 2017
Line 1:		Line 1:
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[Limits_based_on_Mandelstam_Variables\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Initial_Lab_Frame_4-momentum_components\|<math>\triangle </math>]]
		+	[[Special_Case_of_Equal_Mass_Particles\|<math>\vartriangleright </math>]]
		+
		+	</center>
		+
		+
		+
	=Special Case of Equal Mass Particles=		=Special Case of Equal Mass Particles=
	For incoming electrons moving only in the z-direction, we can write		For incoming electrons moving only in the z-direction, we can write

@@ Line 16: / Line 16: @@
 Without knowing the values for gamma or beta, we can utalize the fact that lengths of the two 4-momenta are invariant
-<center><math>s={\mathbf P^*}^2=(E^*_{1}+E^*_{2})^2-(\vec p\ ^*_{1}+\vec p\ ^*_{2})^2=(m_{1}^*+m_{2}^*)^2</math></center>
+<center><math>{\mathbf P^*}^2=(E^*_{1}+E^*_{2})^2-(\vec p\ ^*_{1}+\vec p\ ^*_{2})^2=(m_{1}^*+m_{2}^*)^2</math></center>
-<center><math>s={\mathbf P}^2=(E_{1}+E_{2})^2-(\vec p_{1}+\vec p_{2})^2=(m_{1}+m_{2})^2</math></center>
+<center><math>{\mathbf P}^2=(E_{1}+E_{2})^2-(\vec p_{1}+\vec p_{2})^2=(m_{1}+m_{2})^2</math></center>