Elements of Hamiltonian Mechanics and Electromagnetism

H F Jones

We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the x and p appearing in the Hamiltonian, but quantities X and P constructed by means of the metric operator. Following the analogous procedure of gauging a global symmetry in Hermitian quantum mechanics we find that the corresponding gauge transformation in X implies minimal substitution in the form P → P − e A ( X ). We discuss how the relevant matrix elements governing electromagnetic transitions may be calculated in the special case of the Swanson Hamiltonian, where the equivalent Hermitian Hamiltonian h is local, and in the more generic example of the imaginary cubic interaction, where H is local but h is not.

Iu. G. Pavlenko

Jaroslav Zamastil, Jakub Benda

Timothy B. Boykin

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.

H. F. Jones

We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the x and p appearing in the Hamiltonian, but quantities X and P constructed by means of the metric operator. Following the analogous procedure of gauging a global symmetry in Hermitian quantum mechanics we find that the corresponding gauge transformation in X implies minimal substitution in the form P → P −eA(X). We discuss how the relevant matrix elements governing electromagnetic transitions may be calculated in the special case of the Swanson Hamiltonian, where the equivalent Hermitian Hamiltonian h is local, and in the more generic example of the imaginary cubic interaction, where H is local but h is not. PACS numbers: 03.65.Ca, 11.30.Er, 02.30.Mv 1

Maurice A. de Gosson