A two-dimensional square Bravais lattice has lattice constant

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

Solid State Physics, Ch Bloch's theorem and the nearly-free-electron method

Problem

A two-dimensional square Bravais lattice has lattice constant a=3.2 A˚a=3.2\ \mathrm{\AA}, with lattice vectors that are integer combinations of ax^a\hat{\mathbf x} and ay^a\hat{\mathbf y}. Define the translation operator by TRψ(r)=ψ(r+R)T_{\mathbf R}\psi(\mathbf r)=\psi(\mathbf r+\mathbf R) for any lattice vector R\mathbf R. Take the two lattice vectors RA=2ax^+ay^\mathbf R_A=2a\hat{\mathbf x}+a\hat{\mathbf y} and RB=ax^−ay^\mathbf R_B=a\hat{\mathbf x}-a\hat{\mathbf y}. ((a)) Show that TRATRBψ(r)=TRBTRAψ(r)T_{\mathbf R_A}T_{\mathbf R_B}\psi(\mathbf r)=T_{\mathbf R_B}T_{\mathbf R_A}\psi(\mathbf r) for every function ψ\psi, and give the single translation this product equals, with its magnitude in A˚\mathrm{\AA}. ((b)) The crystal Hamiltonian is H=−ℏ22m∇2+U(r)H=-\dfrac{\hbar^2}{2m}\nabla^2+U(\mathbf r) with U(r+R)=U(r)U(\mathbf r+\mathbf R)=U(\mathbf r) for every lattice vector R\mathbf R. Show that TRAT_{\mathbf R_A} commutes with HH.

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