An electron in a crystal is described by the one-electron Hamiltonian

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

An electron in a crystal is described by the one-electron Hamiltonian H=−ℏ22m∇2+U(r)H=-\dfrac{\hbar^2}{2m}\nabla^2+U(\mathbf r), where the potential is invariant under every Bravais lattice translation, U(r+R)=U(r)U(\mathbf r+\mathbf R)=U(\mathbf r) for all R\mathbf R in the Bravais lattice of the crystal. Define the lattice translation operator TRT_{\mathbf R} acting on any function of position by TRf(r)=f(r+R)T_{\mathbf R}f(\mathbf r) = f(\mathbf r+\mathbf R). Assume Born-von Karman periodic boundary conditions on a large but finite crystal, so every eigenstate can be normalised to unity inside the crystal. ((a)) Show that [TR,H]=0[T_{\mathbf R},H]=0 for every Bravais lattice vector R\mathbf R. ((b)) Show that [TR,TR′]=0[T_{\mathbf R},T_{\mathbf R'}]=0 for every pair of Bravais lattice vectors R,R′\mathbf R,\mathbf R', and that TRTR′=TR+R′T_{\mathbf R}T_{\mathbf R'}=T_{\mathbf R+\mathbf R'}. ((c)) Explain why (a) and (b) allow the eigenstates of HH to be chosen as simultaneous eigenstates of every TRT_{\mathbf R}, with eigenvalue c(R)c(\mathbf R). ((d)) Using the group property from (b) and ∣c(R)∣=1|c(\mathbf R)|=1 (unitarity plus normalisation), derive the form of c(R)c(\mathbf R) and hence Bloch's theorem, ψnk(r)=eik⋅runk(r)\psi_{n\mathbf k}(\mathbf r) = e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r) with unk(r+R)=unk(r)u_{n\mathbf k}(\mathbf r+\mathbf R)=u_{n\mathbf k}(\mathbf r).

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