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5. Describe qualitatively the physical origin of electronic energy bands in solids! a. Show that the spatial probability to find an electron in a Bloch state has the same periodicity as the lattice. b. Show, in one dimension, that the average velocity of Bloch electrons in a band is zero! c. Consider a one dimensional band of the form E(k)= -U cos(kd). Calculate the velocity and effective mass for all k. Consider an electron in this band which is subjected to a constant electric field E. Calculate its position as a function of time, neglecting scattering (the surprising result is called Bloch oscillations and can actually be observed in semiconductor superlattices and ultracold atoms)

5. Describe qualitatively the physical origin of electronic energy bands in solids! a. Show that the spatial probability to find an electron in a Bloch state has the same periodicity as the lattice. b. Show, in one dimension, that the average velocity of Bloch electrons in a band is zero! c. Consider a one dimensional band of the form E(k)= -U cos(kd). Calculate the velocity and effective mass for all k. Consider an electron in this band which is subjected to a constant electric field E. Calculate its position as a function of time, neglecting scattering (the surprising result is called Bloch oscillations and can actually be observed in semiconductor superlattices and ultracold atoms)

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5. Describe qualitatively the physical origin of electronic energy bands in solids! a. Show that the spatial probability to find an electron in a Bloch state has the same periodicity as the lattice. b. Show, in one dimension, that the average velocity of Bloch electrons in a band is zero! c. Consider a one dimensional band of the form E(k)= -U cos(kd). Calculate the velocity and effective mass for all k. Consider an electron in this band which is subjected to a constant electric field E. Calculate its position as a function of time, neglecting scattering (the surprising result is called Bloch oscillations and can actually be observed in semiconductor superlattices and ultracold atoms) d. How many electrons fit into one band assuming spin degeneracy? e. Are ionic solids good or bad conductors? Why?
Transcribed Image Text:5. Describe qualitatively the physical origin of electronic energy bands in solids! a. Show that the spatial probability to find an electron in a Bloch state has the same periodicity as the lattice. b. Show, in one dimension, that the average velocity of Bloch electrons in a band is zero! c. Consider a one dimensional band of the form E(k)= -U cos(kd). Calculate the velocity and effective mass for all k. Consider an electron in this band which is subjected to a constant electric field E. Calculate its position as a function of time, neglecting scattering (the surprising result is called Bloch oscillations and can actually be observed in semiconductor superlattices and ultracold atoms) d. How many electrons fit into one band assuming spin degeneracy? e. Are ionic solids good or bad conductors? Why?
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