For a lattice of spacing

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure · Free electrons, energy bands and electronic structure

Condensed Matter Physics, Ch Electrons in a periodic potential: Bloch's theorem and the opening of a gap

Problem

For a lattice of spacing aa, decide whether each of these waves has Bloch form, and for each one that does, give the wavevector that labels it. Justify each answer by checking whether moving one spacing along, x→x+ax\to x+a, multiplies the whole wave by one constant phase. ((a)) ψ1(x)=eikx[3+cos⁡(4πx/a)]\psi_1(x) = e^{ikx}\left[3+\cos(4\pi x/a)\right] ((b)) ψ2(x)=eikxcos⁡(3πx/a)\psi_2(x) = e^{ikx}\cos(3\pi x/a)

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