Explaining Dispersion Relation for Free Electron - Jayse

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SUMMARY

The discussion focuses on the concept of dispersion relation, specifically for free electrons. It defines the dispersion relation as the relationship between wave number (k) and angular frequency (ω), expressed as k(ω). For free electrons, the relation connects momentum (p) to energy (E) using the equation p = √(E² - m²), where m represents the mass of the electron. Additionally, it highlights the use of this relation in the electron wave function, indicating that the group velocity is given by the formula dE/dp = p/E, assuming the speed of light (c) equals 1.

PREREQUISITES
  • Understanding of wave theory and its terminology
  • Familiarity with special relativity (SR) concepts
  • Knowledge of quantum mechanics, particularly electron wave functions
  • Basic grasp of mathematical relations involving energy and momentum
NEXT STEPS
  • Study the mathematical derivation of dispersion relations in wave mechanics
  • Explore the implications of special relativity on particle physics
  • Learn about the role of wave functions in quantum mechanics
  • Investigate the application of group velocity in different physical contexts
USEFUL FOR

Students and professionals in physics, particularly those studying quantum mechanics, wave theory, and special relativity, will benefit from this discussion.

Jayse_83
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Hi, please could someone explain the term dispersion relation to me, particularly for the case of a free electron?

Thanks

Jayse
 
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In wave theory, a dispersion relation is the relation between the wave number and the angular frequency: [tex]k(\omega)[/tex]. There are also integral relations in S matrix theory that are an integral representation of this. For a free electron, the dispersion relation you refer to must be relating p to E since each is just [tex]\hbar[/tex] times k and [tex]\omega[/tex]. In SR, this is [tex]p=\sqrt{E^2-m^2}[/tex]. It could be used in the electron wave function, so that the group velocity would be
dE/dp=p/E (all with c=1).
 

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