Flux of Incoming Particles - Revision history

Vanwdani: /* Classically */

2018-12-29T20:32:49Z

Classically

2018-12-29T20:32:23Z

Flux of Incoming Particles

2018-12-29T20:31:46Z

2017-07-10T20:51:45Z

Relativistic Quantum Mechanics

2017-07-10T20:36:34Z

Relativistic Quantum Mechanics

2017-07-10T20:22:40Z

Relativistic Quantum Mechanics

2017-07-10T20:20:10Z

Relativistic Quantum Mechanics

2017-07-10T20:19:17Z

Relativistic Quantum Mechanics

2017-07-10T20:13:50Z

Relativistic Quantum Mechanics

2017-07-10T20:13:40Z

Relativistic Quantum Mechanics

← Older revision		Revision as of 20:32, 29 December 2018
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	Beam Power =<math>\le 5000W\ (450A\ at\ 11GeV)</math>		Beam Power =<math>\le 5000W\ (450A\ at\ 11GeV)</math>
−
−	~~----~~

	=Relativistic Quantum Mechanics=		=Relativistic Quantum Mechanics=

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 </center>
-=Flux of Incoming Particles=
-==Classically==
+=Classically=
 Working with a beam of electrons, we can define a pulse as n charges per unit volume, each carrying charge q and moving with speed <math>v_d</math>.  Defining <math>v_d</math> to be the time-average speed of the electrons, we can find the number of particles within a beam.  If A is the beam cross-sectional area, then l, the length of the pulse , has a volume Al with nAl charges within its bounds.  Since each electron carries charge q, the total charge is
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 ----
-==Relativistic Quantum Mechanics==
+=Relativistic Quantum Mechanics=
 <center><math>E^2 \equiv p^2+m^2</math></center>

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 =Flux of Incoming Particles=
 ==Classically==
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 <center><math>F_{cms}=4 \sqrt {\vec p_1^*\,^2s}</math></center>

← Older revision		Revision as of 20:51, 10 July 2017
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−	<center><math>\rho \~~equivi~~(-2iE)\|N\|^2=2E\|N\|^2</math></center>	+	<center><math>\rho \equiv i(-2iE)\|N\|^2=2E\|N\|^2</math></center>

← Older revision		Revision as of 20:36, 10 July 2017
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−	<center><math>\vec j=-i(2i\vec p)N^2=N^2 2p</math></center>	+	<center><math>\vec j \equiv -i(2i\vec p)N^2=N^2 2p</math></center>


		+	and

−	<center><math>\rho=i(-2iE)\|N\|^2=2E\|N\|^2</math></center>	+	<center><math>\rho=i \left (\phi^* \frac{\partial}{\partial t}\phi-\phi \frac{\partial }{\partial t}\phi^* \right )</math></center>
		+
		+
		+	<center><math>\rho=i \left (Ne^{-i\vec p \cdot \vec x +iEt} \frac{\partial}{\partial t}Ne^{i\vec p \cdot \vec x -iEt}-Ne^{i\vec p \cdot \vec x -iEt} \frac{\partial }{\partial t}Ne^{-i\vec p \cdot \vec x +iEt} \right )</math></center>
		+
		+
		+
		+	<center><math>\rho=i \left (-Ne^{-i\vec p \cdot \vec x +iEt}iENe^{i\vec p \cdot \vec x -iEt}-Ne^{i\vec p \cdot \vec x -iEt} iENe^{-i\vec p \cdot \vec x +iEt} \right )</math></center>
		+
		+
		+	<center><math>\rho \equivi(-2iE)\|N\|^2=2E\|N\|^2</math></center>


−	~~<center><math>\vec j=-i(2i\vec p)\|N\|^2=2\vec p\|N\|^2</math></center>~~

← Older revision		Revision as of 20:22, 10 July 2017
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	<center><math>\vec j=-i \left (Ne^{-i\vec p \cdot \vec x +iEt} \frac{\partial}{\partial z} Ne^{i\vec p \cdot \vec x -iEt}-Ne^{i\vec p \cdot \vec x -iEt}\frac{\partial}{\partial z}Ne^{-i\vec p \cdot \vec x +iEt} \right )</math></center>		<center><math>\vec j=-i \left (Ne^{-i\vec p \cdot \vec x +iEt} \frac{\partial}{\partial z} Ne^{i\vec p \cdot \vec x -iEt}-Ne^{i\vec p \cdot \vec x -iEt}\frac{\partial}{\partial z}Ne^{-i\vec p \cdot \vec x +iEt} \right )</math></center>
		+
		+
		+
		+	<center><math>\vec j=-i \left (Ne^{-i\vec p \cdot \vec x +iEt}i\vec p Ne^{i\vec p \cdot \vec x -iEt}+Ne^{i\vec p \cdot \vec x -iEt}i\vec pNe^{-i\vec p \cdot \vec x +iEt} \right )</math></center>
		+
		+
		+
		+	<center><math>\vec j=-i(2i\vec p)N^2=N^2 2p</math></center>

← Older revision		Revision as of 20:20, 10 July 2017
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−	<center><math>\vec j=-i \left (Ne^{-i\vec p \cdot \vec x +iEt} \frac{\partial}{\partial z} Ne^{i\vec p \cdot \vec x -iEt}}-Ne^{i\vec p \cdot \vec x -iEt}\frac{\partial}{\partial z}Ne^{-i\vec p \cdot \vec x +iEt} \right )</math></center>	+	<center><math>\vec j=-i \left (Ne^{-i\vec p \cdot \vec x +iEt} \frac{\partial}{\partial z} Ne^{i\vec p \cdot \vec x -iEt}-Ne^{i\vec p \cdot \vec x -iEt}\frac{\partial}{\partial z}Ne^{-i\vec p \cdot \vec x +iEt} \right )</math></center>

← Older revision		Revision as of 20:19, 10 July 2017
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	where		where

−	~~<center><math>\rho \equiv \Psi^* \Psi </math> is the probability density</center>~~

		+	<center><math>\phi=Ne^{i\vec p \cdot \vec x -iEt}</math></center>

−	~~<center><math>\vec{j} \equiv \rho \vec{v}=\rho \frac{\vec{p}}{m}</math> is the probability current</center>~~	+	This gives

		+	<center><math>\vec j=-i \left (\phi^* \nabla \phi -\phi \nabla \phi^* \right )</math></center>


		+	<center><math>\vec j=-i \left (Ne^{-i\vec p \cdot \vec x +iEt} \frac{\partial}{\partial z} Ne^{i\vec p \cdot \vec x -iEt}}-Ne^{i\vec p \cdot \vec x -iEt}\frac{\partial}{\partial z}Ne^{-i\vec p \cdot \vec x +iEt} \right )</math></center>

−	~~<center><math>\phi=Ne^{i\vec p \cdot \vec x -iEt}</math></center>~~

← Older revision		Revision as of 20:13, 10 July 2017
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−	<center><math>\frac{\partial}{\partial t} \underbrace{ \left [i \left (\phi^\frac{\partial \phi}{\partial t}-\phi \frac{\partial \phi^}{\partial t} \right ) \right ] }_{\rho}+\vec \nabla \cdot \underbrace{ \left [-i\left (\phi^\vec \nabla \phi-\phi \vec \nabla \phi^ \right ) \right ] }_{\~~vec_j~~}=0</math></center>	+	<center><math>\frac{\partial}{\partial t} \underbrace{ \left [i \left (\phi^\frac{\partial \phi}{\partial t}-\phi \frac{\partial \phi^}{\partial t} \right ) \right ] }_{\rho}+\vec \nabla \cdot \underbrace{ \left [-i\left (\phi^\vec \nabla \phi-\phi \vec \nabla \phi^ \right ) \right ] }_{\vec j}=0</math></center>