A spinless particle of mass moves in one dimension in a real

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure

The Structure of Quantum Mechanics, Ch Symmetries, generators and conservation laws

Problem

A spinless particle of mass mm moves in one dimension in a real, time-independent periodic potential V(x+d)=V(x)V(x+d)=V(x), with H^=p^22m+V(x^)\hat H=\dfrac{\hat p^2}{2m}+V(\hat x). Bloch's theorem gives energy eigenstates ψk(x)=eikxuk(x)\psi_k(x)=e^{ikx}u_k(x) with uk(x+d)=uk(x)u_k(x+d)=u_k(x) and energy E(k)E(k). For a spinless particle, time reversal is Θ^=K\hat\Theta=K, complex conjugation of the position-space wave function: Θ^ψ(x)=ψ(x)∗\hat\Theta\psi(x)=\psi(x)^*. (a) Show that [H^,Θ^]=0[\hat H,\hat\Theta]=0 for this H^\hat H. (b) Show that Θ^ψk(x)=ψk(x)∗\hat\Theta\psi_k(x)=\psi_k(x)^* is an eigenstate of H^\hat H with the same energy E(k)E(k), and that it is a Bloch state with wave vector −k-k. (c) Deduce that E(−k)=E(k)E(-k)=E(k) for every band, for any real periodic potential, whether or not VV has any further symmetry (such as being an even function).

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