﻿ Energy-Momentum Conservation | Primer on Geometric Algebra | David Hestenes

## Energy-Momentum Conservation

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The total proper momentum P for an isolated system of particles is conserved:

$\Large{P\,\,\, = \,\,\,\sum\limits_{{\rm{before}}} {{P_k}} \,\,\, = \,\,\,\sum\limits_{{\rm{after}}} {{P_k}} }$

Examples:

(1) Compton Effect (photon scattering off a free electron at rest):    γe  →  γe

Conservation:    P1 + P2  =  P3 + P4

Photons:

$\Large{\begin{array}{l}{P_1}\,\widetilde {{P_1}}\,\,\, = \,\,\,0\,\,\, = \,\,\,{P_3}\,\widetilde {{P_3}}\,\,,\,\,\,\,\,\,\,\,\,\,\,\,p\,\,\, = \,\,\,\left| {\bf{p}} \right|\,\,\, = \,\,\,\frac{E}{c}\,\,\, = \,\,\,\frac{{hf}}{c}\,\,\, = \,\,\,\frac{h}{\lambda }\\{P_1}\,\,\, = \,\,\,\frac{{{E_1}}}{c}\,\left( {1\,\, + \,\,{{{\bf{\hat p}}}_1}} \right)\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{P_3}\,\,\, = \,\,\,\frac{{{E_3}}}{c}\,\left( {1\,\, + \,\,{{{\bf{\hat p}}}_3}} \right)\end{array}}$

Electrons:    P2 2  =  m2c2  =  P4 4

Electron initially at rest:    P2  =  E2/c  =  mc

Problem: Determine the shift in photon frequency due to the scattering (Compton's formula).

(
Click here for a solution to this problem.)

(2) Charged Pion decay:

$\Large{{\pi ^ + }\,\, \to \,\,{\mu ^ + }\,\, + \,\,{\nu _\mu }\,\,\,\,\,\,\,\,\,\,\,\,{\rm{or}}\,\,\,\,\,\,\,\,\,\,\,\,{\pi ^ - }\,\, \to \,\,{\mu ^ - }\,\, + \,\,{{\bar \nu }_\mu }}$

Conservation:
P  =  P1 P2

Problem: Calculate, if possible, the energies of the decay products when the pion decays from rest.

(
Click here for a solution to this problem.)

Previous sectionThe Photon.

Universal Laws for Spacetime Physics.