Energy-Momentum Conservation

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Next section: Universal Laws for Spacetime Physics.


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

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


t1 = constant t2 = constant Interaction P Asymptotic Region Asymptotic Region P1 P2 P P6 P5 P P4 P3



Examples:

          (1) Compton Effect (photon scattering off a free electron at rest):    γ + e−  →  γ + e− 

               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 P̃2  =  m2c2  =  P4 P̃4


               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 section: The Photon.

Next section: Universal Laws for Spacetime Physics.


© David Hestenes 2005, 2014