Tight-binding-like expressions for the continuous-space electromagnetic coupling Hamiltonian

Abstract

In quantum mechanics texts one sometimes encounters the unqualified (and generally untrue) assertion that the solution of the Schrödinger equation for a charged particle in the presence of an externally applied vector potential can be found from that in the absence of the vector potential simply via multiplication with an r-dependent phase factor. The confusion caused by this assertion is only increased when one examines the expression for the matrix elements of the Hamiltonian including electromagnetic interactions (via the vector potential) appropriate for a tight-binding model, for this expression indeed takes a different form from that of the usual minimal coupling Hamiltonian in continuous space. Motivated by the tight-binding (i.e., discretized) result, we derive its continuous-space analog. We show that the aforementioned perplexing assertion actually arises from a confusion between the wavefunction and the matrix elements of the continuous-space Hamiltonian in the position basis and discuss both the tight-binding and continuous-space expressions.