Recursive diagonalization of quantum Hamiltonians to all orders inℏ

Abstract

We present a diagonalization method for generic matrix valued Hamiltonians based on a formal expansion in power of . Considering as a running parameter, a differential equation connecting two diagonalization processes for two very close values of is derived. The integration of this differential equation allows the recursive determination of the series expansion in powers of for the diagonalized Hamiltonian. This approach results in effective Hamiltonians with Berry phase corrections of higher order in , and deepens previous works on the semiclassical diagonalization of quantum Hamiltonians which led notably to the discovery of the intrinsic spin Hall effect. As physical applications we consider spinning massless particles in isotropic inhomogeneous media and show that both the energy and the velocity get quantum corrections of order 2 . We also derive formally to all order in the energy spectrum and the equations of motion of Bloch electrons in an external electric 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.…”

“…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.…”

“…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.…”