How Do Phase and Group Velocities Behave in Relativistic Electron Waves?

[Mindscrape]

The dispersion relation for the free relativisitic electron wave is $$\omega (k) = \sqrt{c^2 k^2 + (m_e c^2/ \hbar)^2}$$. Obtain expressions for the phase velocity and group velocity of these waves and show that their product is a constant, independent of k. From your result, what can you conclude about the group velocity if the phase velocity is greater than the speed of light?

The group velocity will be easy to find because I can just differentiate with respect to k. I am not really sure what to do for the phase velocity. I figure that since $$v_p = f \lambda = E/p$$ then I could use the relativistic energy expression $$E = (p^2 c^2 + m^2 c^4)^{\frac{1}{2}}$$. I am unsure about how to tackle the momentum. Does an electron have a de Broglie wave dispersion?

 

[OlderDan]

Showing the product to be constant was just done in another thread, but you can do it on your own. Phase velocity is ω/k.

 

[kyle8921]

I can provide a response to this content by providing some clarifications and additional information.

Firstly, the dispersion relation given in the content is for a free relativistic electron, meaning that it is not affected by any external forces or fields. This is an important distinction to make, as the dispersion relation for a relativistic electron in a potential is different.

To obtain the expressions for the phase velocity and group velocity, we can use the definitions v_p = \frac{\omega}{k} and v_g = \frac{d\omega}{dk}, respectively. Plugging in the dispersion relation, we get v_p = \frac{c^2 k}{\sqrt{c^2 k^2 + (m_e c^2/\hbar)^2}} and v_g = \frac{c^2}{\sqrt{c^2 k^2 + (m_e c^2/\hbar)^2}}.

To show that their product is a constant, we can simply calculate v_p \cdot v_g and see that it is indeed equal to c^2, independent of k. This is a result of the relativistic energy-momentum relation, which is used in the derivation of the dispersion relation.

From this result, we can conclude that the group velocity is always less than or equal to the speed of light, regardless of the phase velocity. This is a fundamental property of relativistic particles, as they cannot travel faster than the speed of light. So, if the phase velocity is greater than the speed of light, the group velocity will still be limited to c. This is a manifestation of the time dilation and length contraction effects predicted by special relativity.

In terms of the de Broglie wave dispersion, it is indeed applicable to electrons, as they exhibit wave-like behavior. However, the dispersion relation for a free relativistic electron is different from the one used for non-relativistic particles. This is because the relativistic energy-momentum relation is different from the classical one.

In conclusion, the dispersion relation for a free relativistic electron can provide valuable insights into the behavior of these particles at high energies and speeds. The expressions for phase velocity and group velocity can also be useful in understanding the propagation of these waves.