Total Energy in CM Frame - Revision history

Vanwdani at 18:54, 15 May 2018

2018-05-15T18:54:14Z

Vanwdani at 01:08, 16 June 2017

2017-06-16T01:08:56Z

Vanwdani at 22:34, 15 June 2017

2017-06-15T22:34:35Z

Vanwdani at 22:29, 15 June 2017

2017-06-15T22:29:45Z

Vanwdani at 22:10, 15 June 2017

2017-06-15T22:10:51Z

Vanwdani: Created page with "=Total Energy in CM Frame= Setting the lengths of the 4-momenta equal to each other, ${\mathbf P^*}^2={\mathbf P}^2$ we can use this for the col…"

2017-05-31T17:58:24Z

Created page with "=Total Energy in CM Frame= Setting the lengths of the 4-momenta equal to each other, <center><math>{\mathbf P^*}^2={\mathbf P}^2</math></center> we can use this for the col…"

New page

=Total Energy in CM Frame=

Setting the lengths of the 4-momenta equal to each other,

<center><math>{\mathbf P^*}^2={\mathbf P}^2</math></center>

we can use this for the collision of two particles of mass m. Since the total momentum is zero in the Center of Mass frame, we can express total energy in the center of mass frame as

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

<center><math>(E^*)^2-(\vec p\ ^*)^2=(E_{1}+E_{2})^2-(\vec{p_{1}}+\vec p_{2})^2</math></center>

<center><math>(E^*)^2=(E_{1}+E_{2})^2-(\vec{p_{1}}+\vec p_{2})^2</math></center>

<center><math>E^*=\sqrt{(E_{1}+E_{2})^2-(\vec{p_{1}}+\vec p_{2})^2}</math></center>

<center><math>E^*=\sqrt{E_{1}^2+2E_{1}E_{2}+E_{2}^2-\vec p_{1} . \vec p_{2} -\vec p_{1} . \vec p_{1} -\vec p_{2} . \vec p_{1} -\vec p_{2} . \vec p_{2} }</math></center>

<center><math>E^*=\sqrt{(E_{1}^2- p_{1}^2 )+(E_{2}^2-p_{2}^2 )+2E_{1}E_{2}-\vec p_{1} . \vec p_{2} -\vec p_{2} . \vec p_{1} }</math></center>

<center><math>E^*=\sqrt{(E_{1}^2- p_{1}^2 )+(E_{2}^2-p_{2}^2 )+2E_{1}E_{2}-p_{1} p_{2}\cos(\theta) - p_{2} p_{1}\cos(\theta) }</math></center>

<center><math>E^*=\sqrt{m_{1}^2+m_{2}^2+2E_{1}E_{2}-p_{1} p_{2}\cos(\theta) - p_{2} p_{1}\cos(\theta) }</math></center>

<center><math>E^*=\sqrt{m_{1}^2+m_{2}^2+2E_{1}E_{2}-2p_{1} p_{2}\cos(\theta) }</math></center>

<center><math>E^*=\sqrt{m_{1}^2+m_{2}^2+2E_{1}E_{2}-2p_{1} p_{2}\cos(\theta) }</math></center>

Using the relations <math>\beta\equiv \vec p/E\Longrightarrow \vec p=\beta E</math>

<center><math>E^*=\sqrt{2m^2+2E_{1}E_{2}(1-\beta_{1}\beta_{2}\cos(\theta))}</math></center>

<center>where <math> \theta </math> is the angle between the particles in the Lab frame.</center>

In the frame where one particle ''(p<sub>2</sub>)'' is at rest

<center><math>\Longrightarrow \beta_{2}=0</math></center>

<center><math>\Longrightarrow p_{2}=0</math></center>

which implies,

<center><math> E_{2}=\sqrt{p_{2}^2+m^2}=m</math></center>

{| class="wikitable" align="center"
| style="background: gray" | <math>E^*=(2m(m+E_{1})^{1/2}=(2(.511MeV)(.511MeV+\sqrt{(11000 MeV)^2+(.511 MeV)^2})^{1/2}\approx 106.030760886 MeV</math>
|}

where <math>E_{1}=\sqrt{p_{1}^2+m^2}\approx 11000 MeV</math>

@@ Line 1: / Line 1: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Special_Case_of_Equal_Mass_Particles|<math>\vartriangleleft </math>]]
@@ Line 90: / Line 90: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Special_Case_of_Equal_Mass_Particles|<math>\vartriangleleft </math>]]

← Older revision		Revision as of 01:08, 16 June 2017
Line 83:		Line 83:

	where <math>E_{1}=\sqrt{p_{1}^2+m^2}\approx 11000 MeV</math>		where <math>E_{1}=\sqrt{p_{1}^2+m^2}\approx 11000 MeV</math>
		+
		+
		+
		+	----
		+
		+
		+
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[Special_Case_of_Equal_Mass_Particles\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Initial_4-momentum_Components\|<math>\triangle </math>]]
		+	[[Scattered_and_Moller_Electron_Energies_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>]]
+[[Special_Case_of_Equal_Mass_Particles|<math>\vartriangleleft </math>]]
 [[VanWasshenova_Thesis#Initial_4-momentum_Components|<math>\triangle </math>]]
-[[Special_Case_of_Equal_Mass_Particles|<math>\vartriangleright </math>]]
+[[Scattered_and_Moller_Electron_Energies_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 22:10, 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>
		+
		+
	=Total Energy in CM Frame=		=Total Energy in CM Frame=