Final CM Frame Moller Electron 4-momentum components - Revision history

Vanwdani: /* Final CM Frame Moller Electron 4-momentum components */

2018-02-26T18:20:29Z

Final CM Frame Moller Electron 4-momentum components

Vanwdani: /* Final CM Frame Moller Electron 4-momentum components */

2018-02-26T18:12:26Z

Final CM Frame Moller Electron 4-momentum components

Vanwdani at 03:02, 16 June 2017

2017-06-16T03:02:13Z

Vanwdani: /* Final CM Frame Moller Electron 4-momentum components */

2017-06-16T03:00:13Z

Final CM Frame Moller Electron 4-momentum components

Vanwdani: /* Final CM Frame Moller Electron 4-momentum components */

2017-06-16T02:59:51Z

Final CM Frame Moller Electron 4-momentum components

Vanwdani: /* Final CM Frame Moller Electron 4-momentum components */

2017-06-16T02:58:49Z

Final CM Frame Moller Electron 4-momentum components

Vanwdani: /* Final CM Frame Moller Electron 4-momentum components */

2017-06-16T02:51:47Z

Final CM Frame Moller Electron 4-momentum components

Vanwdani at 22:16, 15 June 2017

2017-06-15T22:16:42Z

Vanwdani: Created page with "=Final CM Frame Moller Electron 4-momentum components= Relativistically, the x and y components remain the same in the conversion from the Lab frame to the Center of Mass frame,…"

2017-05-31T19:40:32Z

Created page with "=Final CM Frame Moller Electron 4-momentum components= Relativistically, the x and y components remain the same in the conversion from the Lab frame to the Center of Mass frame,…"

New page

=Final CM Frame Moller Electron 4-momentum components=

Relativistically, the x and y components remain the same in the conversion from the Lab frame to the Center of Mass frame, since the direction of motion is only in the z direction.

{| class="wikitable" align="center"
| style="background: gray" | <math>p^*_{2(x)}\Leftrightarrow p_{2(x)}'</math>
|}

{| class="wikitable" align="center"
| style="background: gray" | <math>p^*_{2(y)}\Leftrightarrow p_{2(y)}'</math>
|}

{| class="wikitable" align="center"
| style="background: gray" | <math>p^*_{2(z)}=-\sqrt {(p^*_2)^2-(p^*_{2(x)})^2-(p^*_{2(y)})^2}</math>
|}

We choose negative, since the incoming electron in the lab frame is traveling in the positive direction, and the Moller electron is initially at rest, which translates to negative motion in the CM frame.

Redefining the components in simpler terms, we use the fact that
<center><math>E^*\equiv E^*_1+E^*_2</math></center>

<center><math>2E^*_2=\sqrt{2m(m+E_1)}</math></center>

{| class="wikitable" align="center"
| style="background: gray" | <center><math>E^*_2=\sqrt{\frac{m(m+E_1)}{2}}</math></center>
|}

{| class="wikitable" align="center"
| style="background: gray" | <math> p^*_{2}=\sqrt{E_{2}^{*2}-m^2}</math>
|}
Initially, before the collision in the CM frame, p<sub>2</sub> was in the negative z direction. After the collision, the direction should reverse to the positive z direction. This same switching of the momentum direction alters p<sub>1</sub> as well.

<center>Using <math>\theta '_2=\arccos \left(\frac{p^'_{2(z)}}{p^'_{2}}\right)</math></center>

{| class="wikitable" align="center"
| style="background: gray" | <math>\Longrightarrow \theta ^*_2=\arccos \left(\frac{p^*_{2(z)}}{p^*_{2}}\right)</math>
|}

@@ Line 73: / Line 73: @@
-For the case where <math>/theta \equal 180^{\circ}</math>
+For the case where <math>\theta = 180^{\circ}</math>
 This implies
-<math>\cos{\frac{p_z}{p}} \equals </math>-1
+<math>\cos{\frac{p_z}{p}} =-1 </math>
 ----

← Older revision		Revision as of 18:12, 26 February 2018
Line 59:		Line 59:


−	~~<center>~~Using <math>\theta '_2=\arccos \left(\frac{p^'_{2(z)}}{p^'_{2}}\right)</math>~~</center>~~	+	Using
		+
		+
		+	<math>\theta '_2=\arccos \left(\frac{p^'_{2(z)}}{p^'_{2}}\right)</math>
		+
		+
		+

	{\| class="wikitable" align="center"		{\| class="wikitable" align="center"
Line 66:		Line 72:


		+
		+	For the case where <math>/theta \equal 180^{\circ}</math>
		+
		+	This implies
		+
		+
		+	<math>\cos{\frac{p_z}{p}} \equals </math>-1

@@ Line 1: / Line 1: @@
 <center><math>\textbf{\underline{Navigation}}</math>
-[[Limits_based_on_Mandelstam_Variables|<math>\vartriangleleft </math>]]
+[[Momentum_Components_in_the_XY_Plane_Based_on_Angle_Phi|<math>\vartriangleleft </math>]]
-[[VanWasshenova_Thesis#Final_Lab_Frame_Moller_Electron_4-momentum_components|<math>\triangle </math>]]
+[[VanWasshenova_Thesis#Final_4-momentum_components|<math>\triangle </math>]]
-[[Special_Case_of_Equal_Mass_Particles|<math>\vartriangleright </math>]]
+[[Final_CM_Frame_Scattered_Electron_4-momentum_components|<math>\vartriangleright </math>]]
 </center>
@@ Line 64: / Line 64: @@
 | style="background: gray"  | <math>\Longrightarrow \theta ^*_2=\arccos \left(\frac{p^*_{2(z)}}{p^*_{2}}\right)</math>
 |}

← Older revision		Revision as of 03:00, 16 June 2017
Line 42:		Line 42:


−	Initially <math>\vec p_2=0 \quad \Rightarrow E^2=p^2+m^2=m^2</math>~~</center>~~	+	Initially <math>\vec p_2=0 \quad \Rightarrow E^2=p^2+m^2=m^2</math>
−

← Older revision		Revision as of 02:59, 16 June 2017
Line 38:		Line 38:


−	<center><math>s \equiv (P_1^+P_2^)^2=(E^_1+E^_2)^2=(2E^*_2)^2=(P_1+P_2)^2=2m^2+2E_1E_2-2 \vec p_1 \cdot \vec p_2/math></center>	+	<center><math>s \equiv (P_1^+P_2^)^2=(E^_1+E^_2)^2=(2E^*_2)^2=(P_1+P_2)^2=2m^2+2E_1E_2-2 \vec p_1 \cdot \vec p_2</math></center>

← Older revision		Revision as of 02:58, 16 June 2017
Line 38:		Line 38:


−	<center><math>s \equiv (P_1^+P_2^)^2=(E^_1+E^_2)^2=(2E^*_2)^2=(P_1^'+P_2^')^2</math></center>	+	<center><math>s \equiv (P_1^+P_2^)^2=(E^_1+E^_2)^2=(2E^*_2)^2=(P_1+P_2)^2=2m^2+2E_1E_2-2 \vec p_1 \cdot \vec p_2/math></center>



−	<center><math>2E^*_2=\sqrt{2m(m+E_1)}</math></center>	+	Initially <math>\vec p_2=0 \quad \Rightarrow E^2=p^2+m^2=m^2</math></center>
		+
		+
		+
		+	<center><math>\sqrt{2}E^*_2=\sqrt{2m(m+E_1)}</math></center>

@@ Line 36: / Line 36: @@
 Redefining the components in simpler terms, we use the fact that
 <center><math>E^*\equiv E^*_1+E^*_2</math></center>
 <center><math>2E^*_2=\sqrt{2m(m+E_1)}</math></center>

← Older revision		Revision as of 22:16, 15 June 2017
Line 1:		Line 1:
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[Limits_based_on_Mandelstam_Variables\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Final_Lab_Frame_Moller_Electron_4-momentum_components\|<math>\triangle </math>]]
		+	[[Special_Case_of_Equal_Mass_Particles\|<math>\vartriangleright </math>]]
		+
		+	</center>
		+
		+
	=Final CM Frame Moller Electron 4-momentum components=		=Final CM Frame Moller Electron 4-momentum components=