Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

Abstract

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at orderh for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.

Comparative analysis of direct and “step-by-step” Foldy-Wouthuysen transformation methods

“…We also note that this agreement is an additional confirmation of the correctness of the quantum mechanics formalisms used in [17,[19][20][21][22].…”

“…Although some fairly general problems were considered in [17,[19][20][21][22], the above form was not used there. On the other hand, a concrete form of the Hamiltonian operator in the FW representation was obtained in [3] in the weak-field approximation, i.e., with only first-order terms in field potentials and their derivatives taken into account.…”

“…Quantum mechanical evolution is described using Moyal brackets, corresponding to commutators in ordinary quantum mechanics, while observable quantities are characterized by functions on the phase space. In the original method proposed in [19][20][21][22], unitary transformations are also not used. The quantum mechanical system Hamiltonian is represented as a matrix H 0 (P , R), whose elements are operators depending on the pair of canonical variables P and R. This method determines a diagonalization procedure based on formal series in powers of the Planck constant and can be used for a wide class of Hamiltonians, for which Berry phase corrections are essential.…”

“…But it easily seen that it is problematic to effectively use the Eriksen method with the goal of obtaining relativistic formulas for particles in an external field because general formula (5) In contrast to the Eriksen method, some of the iterative methods give the sought relativistic expressions [3,[17][18][19][20][21][22]. We note that the method developed in [18] is applicable to particles with any spin.…”

“…Here, we compare results obtained by the three methods developed in [17], in [3,18] and in [19][20][21][22]. These methods have the most fundamental justification among the methods of the step-by-step type.…”