A particle of mass moves in the - plane under a potential energy periodic with the…

PHYSICS · Multi-part · 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 particle of mass mm moves in the xx-yy plane under a potential energy V(x,y)V(x,y) periodic with the same period dd along both axes: V(x+d,y)=V(x,y+d)=V(x,y)V(x+d,y)=V(x,y+d)=V(x,y) for all x,yx,y (an idealised infinite two-dimensional square lattice). Let T^x(d)\hat T_x(d) and T^y(d)\hat T_y(d) be translation by dd along xx and along yy, and H^=p^x2+p^y22m+V(x^,y^)\hat H=\dfrac{\hat p_x^2+\hat p_y^2}{2m}+V(\hat x,\hat y). ((a)) Explain why H^\hat H, T^x(d)\hat T_x(d) and T^y(d)\hat T_y(d) are three mutually commuting operators and so share a common eigenbasis. ((b)) Let ∣ψ⟩|\psi\rangle be a simultaneous eigenstate with T^x(d)∣ψ⟩=e−ikxd∣ψ⟩\hat T_x(d)|\psi\rangle=e^{-ik_xd}|\psi\rangle and T^y(d)∣ψ⟩=e−ikyd∣ψ⟩\hat T_y(d)|\psi\rangle=e^{-ik_yd}|\psi\rangle, using the convention (T^x(d)ψ)(x,y)=ψ(x−d,y)(\hat T_x(d)\psi)(x,y)=\psi(x-d,y) of the section on lattice translation. Show that ψ(x,y)=ei(kxx+kyy)u(x,y)\psi(x,y)=e^{i(k_xx+k_yy)}u(x,y) with u(x+d,y)=u(x,y+d)=u(x,y)u(x+d,y)=u(x,y+d)=u(x,y). ((c)) Show that (kx,ky)(k_x,k_y) and (kx+2π/d, ky)(k_x+2\pi/d,\,k_y) label the same state, so kxk_x (and likewise kyk_y) is only defined modulo 2π/d2\pi/d.

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