Question Solved1 Answer (a) Show that Bloch wavefunctions are not eigenfunctions of the momentum operator. (b) According to Ehrenfest's theorem, any operator that commutes with the Hamiltonian shares eigenstates with the Hamiltonian, and is a conserved quantity. Show that the momentum operator does not commute with the Hamiltonian. (c) From your answer to (a) and (b), we may conclude that the momentum of an electron in a periodic potential is not conserved. However, let's define a quantity hk where k is a wavevector. Show that the Bloch wavefunction is unaffected by a change k + k + K, where is any reciprocal lattice vector. The so-called crystal momentum of an electron, hk is defined only up to a reciprocal lattice vector. It is the "conserved" quantity regarding momentum. (d) From the model of a periodic potential that results in Bloch wavefunctions, there would appear to be no change in crystal momentum (up to a reciprocal lattice vector), and therefore no effective scattering. This can be confirmed by deriving the mean velocity, vn(k) of electrons in a band n and with wavevector k. The expression is (see slide 10 of Lecture 5's slides, or Appendix E of Ashcroft and Mermin for a derivation): v (k) - Vken(k) You do not have to derive this expression for this HW. Using the above expression show that, indeed, the mean velocity of the electron in a Bloch state (of fixed band, n) is conserved, and that it is unaffected by a change k→ K+K, where is any reciprocal lattice vector. (e) What we have discussed in this question runs contrary to the Drude model of Lecture 1. What are some of the sources of scattering in a real crystal that we have left out of the quantum mechanical model up to this point?

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Transcribed Image Text: (a) Show that Bloch wavefunctions are not eigenfunctions of the momentum operator. (b) According to Ehrenfest's theorem, any operator that commutes with the Hamiltonian shares eigenstates with the Hamiltonian, and is a conserved quantity. Show that the momentum operator does not commute with the Hamiltonian. (c) From your answer to (a) and (b), we may conclude that the momentum of an electron in a periodic potential is not conserved. However, let's define a quantity hk where k is a wavevector. Show that the Bloch wavefunction is unaffected by a change k + k + K, where is any reciprocal lattice vector. The so-called crystal momentum of an electron, hk is defined only up to a reciprocal lattice vector. It is the "conserved" quantity regarding momentum. (d) From the model of a periodic potential that results in Bloch wavefunctions, there would appear to be no change in crystal momentum (up to a reciprocal lattice vector), and therefore no effective scattering. This can be confirmed by deriving the mean velocity, vn(k) of electrons in a band n and with wavevector k. The expression is (see slide 10 of Lecture 5's slides, or Appendix E of Ashcroft and Mermin for a derivation): v (k) - Vken(k) You do not have to derive this expression for this HW. Using the above expression show that, indeed, the mean velocity of the electron in a Bloch state (of fixed band, n) is conserved, and that it is unaffected by a change k→ K+K, where is any reciprocal lattice vector. (e) What we have discussed in this question runs contrary to the Drude model of Lecture 1. What are some of the sources of scattering in a real crystal that we have left out of the quantum mechanical model up to this point?
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Transcribed Image Text: (a) Show that Bloch wavefunctions are not eigenfunctions of the momentum operator. (b) According to Ehrenfest's theorem, any operator that commutes with the Hamiltonian shares eigenstates with the Hamiltonian, and is a conserved quantity. Show that the momentum operator does not commute with the Hamiltonian. (c) From your answer to (a) and (b), we may conclude that the momentum of an electron in a periodic potential is not conserved. However, let's define a quantity hk where k is a wavevector. Show that the Bloch wavefunction is unaffected by a change k + k + K, where is any reciprocal lattice vector. The so-called crystal momentum of an electron, hk is defined only up to a reciprocal lattice vector. It is the "conserved" quantity regarding momentum. (d) From the model of a periodic potential that results in Bloch wavefunctions, there would appear to be no change in crystal momentum (up to a reciprocal lattice vector), and therefore no effective scattering. This can be confirmed by deriving the mean velocity, vn(k) of electrons in a band n and with wavevector k. The expression is (see slide 10 of Lecture 5's slides, or Appendix E of Ashcroft and Mermin for a derivation): v (k) - Vken(k) You do not have to derive this expression for this HW. Using the above expression show that, indeed, the mean velocity of the electron in a Bloch state (of fixed band, n) is conserved, and that it is unaffected by a change k→ K+K, where is any reciprocal lattice vector. (e) What we have discussed in this question runs contrary to the Drude model of Lecture 1. What are some of the sources of scattering in a real crystal that we have left out of the quantum mechanical model up to this point?
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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