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Mindscrape
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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?
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?
