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" (d) "{:[[v_(1),H]=[(i)/(ℏ)[H,r],H]=0],[v_(n)(k)=(i)/(ℏ)(del)/(del k)E_(n)(k)],[k rarr k+K],[v_(n)(k)=(i)/(ℏ)(del)/(del(k+K))E_(n)(k+k)],[=(i)/(ℏ)(delE_(n)(k))/(del k)]:}:. velocity is unaffected by the change k rarr k+K(a.) Bloch Wavefunction{:[Psi=u_(k)(x)e^(ikx)],[ hat(p)|Psi:)=(-iℏ(d)/(dx))u_(k)(x)e^(ikx)],[=-iℏ[iku_(k)(x)e^(ikx)+u_(k)^(')(x)e^(ikx)]],[=(-pu_(k)(x)-iℏu_(k)^(')(x))e^(ikx)]:}where p=ℏk:. Bloch wavefunctione are not eigenfunctions of momentim operator(b){:[H=(-ℏ^( ... See the full answer