Explaining Dispersion Relation for Free Electron - Jayse

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Dispersion relation describes the relationship between wave number and angular frequency, denoted as k(ω). For a free electron, it relates momentum (p) to energy (E), represented by the equation p = √(E² - m²). This relationship can be integrated into the electron wave function, influencing the group velocity, calculated as dE/dp = p/E. The discussion emphasizes the significance of this relation in understanding wave behavior in quantum mechanics. Overall, the dispersion relation is crucial for analyzing free electron dynamics.
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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: k(\omega). 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 \hbar times k and \omega. In SR, this is p=\sqrt{E^2-m^2}. 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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