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Question: the dispersion relationship wwk is important for example it gives...

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The dispersion relationship w=w(k) is important. For example, it gives the phase and group velocities. w is directly related to the energy, since the reason for the introduction of w is the time evolution of the wavefunction exp(-iEt/ ℏ). The textbook shows you only the non-relativistic case for a particle with rest mass m0, w ≡ E/ ℏ = p^2/(2m0ℏ)=(ℏk)^2/(2m0ℏ) = ℏk^2/2m0. Now, please consider the relativistic case where E=sqrt(p^2c^2+m0^2c^4)=mc^2 , here the 2-nd m is the relativistic mass, not m0. Find the group and phase velocity of a relativistic particle that travels at velocity v in the classical sense. Hint: use textbook equations 2.108 and 2.109, and compare the final results with v. Note also p= mv where m is the relativistic mass

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