Explaining Dispersion Relation for Free Electron - Jayse

[Jayse_83]

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

Thanks

Jayse

 

[Meir Achuz]

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).

 

[sir-pinski]

Hi Jayse,

Sure, I'd be happy to explain the dispersion relation for a free electron.

In physics, the dispersion relation is a mathematical relationship that describes how the energy and momentum of a particle are related. In simpler terms, it shows how the energy of a particle changes as its momentum changes.

For a free electron, the dispersion relation can be described by the equation E = ħ²k²/2m, where E is the energy, ħ is the reduced Planck constant, k is the wave vector (related to the momentum), and m is the mass of the electron.

This means that as the momentum (or wave vector) of a free electron increases, its energy also increases. This relationship is linear, meaning that the energy increases in proportion to the square of the momentum.

In other words, the dispersion relation for a free electron shows that the energy of the electron increases as its momentum increases, but the rate at which it increases is dependent on the mass of the electron. This is why electrons with different masses (such as in different materials) have different dispersion relations.

I hope this helps to clarify the concept of dispersion relation for you. Let me know if you have any further questions.