Diagonal representation for a generic matrix valued quantum Hamiltonian

2008/01/31 by Pierre Gosselin, Herve Mohrbach, Hervé Mohrbach · 3 citations
Mathematics · Physics and Astronomy · #Adiabatic quantum computation #Diagonal #Diagonal matrix #Hamiltonian (control theory) #Matrix representation #Noncommutative and Quantum Gravity Theories #Planck #Planck constant #Power series #Quantum Mechanics and Non-Hermitian Physics #Quantum and Classical Electrodynamics #cond-mat.other #hep-th #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1140/epjc/s10052-009-1155-3

published as Eur.Phys.J.C64:495-527,2009 · Significant revision, typos corrected and references added

arxiv created 2009/05/25 · openalex publication_date 2009/09/29 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

A general method to derive the diagonal representation for a generic matrix valued quantum Hamiltonian is proposed. In this approach new mathematical objects like non-commuting operators evolving with the Planck constant promoted as a running variable are introduced. This method leads to a formal compact expression for the diagonal Hamiltonian which can be expanded in a power series of the Planck constant. In particular, we provide an explicit expression for the diagonal representation of a generic Hamiltonian to the second order in the Planck constant. This last result is applied, as a physical illustration, to Dirac electrons and neutrinos in external fields.

Citations

• Modified geometrical optics of a smoothly inhomogeneous isotropic medium: The anisotropy, Berry phase, and the optical Magnus effect
• Semiclassical dynamics of Dirac particles interacting with a static gravitational field
• Geometrodynamics of spinning light
• Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections
• Recursive diagonalization of quantum Hamiltonians to all orders inℏ
• Topological spin transport of photons: the optical Magnus effect and Berry phase
• Quantal phase factors accompanying adiabatic changes
• Topological spin transport of a relativistic electron
• On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit

Cited by

• General properties of the Foldy-Wouthuysen transformation and applicability of the corrected original Foldy-Wouthuysen method
• Relativistic Quantum Dynamics of Twisted Electron Beams in Arbitrary Electric and Magnetic Fields
• Validation of the Eriksen method for the exact Foldy-Wouthuysen representation