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Phase and Group Velocity for Relativistic Particle

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Physics Modern Physics quantum physics

https://texercises.com/exercise/phase-and-group-velocity-for-relativistic-particle/

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Exercise:
For relativistic particles the total energy is given by the relation E^ pc^+E_^ where p and E_ are the particle's momentum and rest energy respectively. Derive formal expressions for the phase and group velocity and show that their product is equal to the square of the speed of light.

Solution:
The dispersion relation is omegak fracEkhbar fracsqrtpc^+E_^hbar fracsqrthbar k c^ + E_^hbar It follows for the phase velocity v_p fracEhbar k fracEp and for the group velocity v_g fractextdtextdk left fracsqrthbar k c^ + E_^hbarright frachbar c^ khbar sqrthbar k c^ + E_^ frachbar c^ ksqrthbar k c^ + E_^ fracp c^E The product of the phase and group velocity is v_p v_g fracEp fracp c^E c^ Remark: Using the relativistic expressions for energy and momentum E gamma E_ gamma m c^ p gamma m v the group velocity can be written as v_g fracgamma m v c^gamma m c^ v As expected the group velocity corresponds to the velocity of the classical particle.

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Branches	quantum physics
Tags	duality, energy, group velocity, mass, momentum, particle, phase velocity, relativity, wave
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Difficulty	(2, default)
Points	0 (default)
Language	(English)
Type	Algebraic
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