Electrons in a one-dimensional crystal experience the periodic potential

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Position, momentum and the origin of the canonical commutator

Problem

Electrons in a one-dimensional crystal experience the periodic potential V(x)=V0cos⁡ ⁣(2πxd)V(x) = V_0\cos\!\left(\dfrac{2\pi x}{d}\right), with lattice spacing d=0.40 nmd=0.40\,\text{nm}. The Hamiltonian is quantised canonically as H^=p^2/2m+V(x^)\hat H = \hat p^2/2m + V(\hat x); there is no ordering ambiguity here since VV depends only on x^\hat x.A problem set states: "because the crystal potential is periodic, it is invariant under translation. By the translation-invariance argument, a periodic VV therefore commutes with p^\hat p, so [H^,p^]=0[\hat H,\hat p]=0 and the electron's momentum pp is conserved exactly as it would be for a free particle." ((a)) Compute [V^(x^),p^][\hat V(\hat x),\hat p] using ⟨x∣p^∣ψ⟩=−iℏ∂xψ(x)\langle x|\hat p|\psi\rangle=-i\hbar\partial_x\psi(x) and say whether it vanishes for general xx. ((b)) Identify exactly which translations leave VV unchanged, and state what operator's eigenvalue is actually conserved, together with its name.

Related theory

Browse the problem library