An electron near the bottom of a conduction band is often modelled with a…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics
The Structure of Quantum Mechanics, Ch Position, momentum and the origin of the canonical commutator
Problem
An electron near the bottom of a conduction band is often modelled with a position-dependent effective mass m(x), with classical Hamiltonian H=2m(x)p2. Take the simple case m(x)=m0(1+Lx) for a length scale L, and quantise to leading order in x/L with the symmetrised ordering, so that H^≈2m0p^2−4m0L1(x^p^2+p^2x^), which is Hermitian.A study group argues: "the canonical commutator [x^,p^]=iℏ holds no matter how we order x^ and p^ when we build H^, so the ordering choice cannot affect whether p^ is conserved; since [x^,p^] is fixed, [H^,p^]=0 regardless of the ordering, exactly as for a free particle."
((a)) Compute [H^,p^] for the stated ordering and say whether it is zero.
((b)) Explain the logical error in the study group's reasoning, referring to what actually controls whether a Hamiltonian commutes with the generator of translations.