Diagonalization of multicomponent wave equations with a Born-Oppenheimer example

Abstract

Article:Weigert, S. orcid.org/0000-0002-6647-3252 and Littlejohn, Robert (1993 ReuseUnless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. A general method to decouple multicomponent linear wave equations is presented. First, the Weyl calculus is used to transform operator relations into relations between c-number valued matrices. Then it is shown that the symbol representing the wave operator can be diagonalized systematically up to arbitrary order in an appropriate expansion parameter. After transforming the syn~bols back to operators, the original problem is reduced to solving a set of linear uncoupled scalar wave equations. The procedure is exemplified for a particle with a Born-Oppenheimer-type Hamiltonian valid through second order in fi. The resulting effective scalar Harniltonians are seen to contain an additional velocitydependent potential. This contribution has not been reported in recent studies investigating the adiabatic motion of a neutral particle moving in an inhomogeneous magnetic field. Finally, the relation of the general method to standard quantum-mechanical perturbation theory is discussed. PACS numberk): 03.65.Sq, 03.40.Kf

Diagonal representation for a generic matrix valued quantum Hamiltonian

“…It has also been understood, in the context of the Born-Oppenheimer theory of molecules [2][3] [4] that the geometric phases induced by the quantum part of the system provide a correction, reacting on the classical part with geometric Lorentz and electric forces [5] [6]. Going beyond this first correction has proved extraordinarily difficult due to the intricate entanglement of noncommuting operators [7].…”

“…Among other approaches, Weigert and Littlejohn developed a systematic method to diagonalize general quantum Hamiltonian in a series expansion in [7]. It leads also to a formal series expansion written in terms of symbols of operators which also makes the method complicated for practical applications.…”

Adiabatic motion of a neutral spinning particle in an inhomogeneous magnetic field

“…(7). In the first equation, the force V(B.s) on the particle is decomposed into two parts, the first of which, (b-s)VB, is due to the changing strength of the magnetic field, whereas the second one, BVb.s,=BVb-S, originates from the directional change of the field from point to point.…”

“…Next, if the particle moves in the time dt from the point X to the point X', the change dm of the variable m is due solely to the change in direction of the field B, which implies the second of Eqs. (7).…”

The time-dependent Born-Oppenheimer approximation

“…In the physics literature the correct second order Born-Oppenheimer Hamiltonian (48) was first obtained by Weigert and Littlejohn in [31] for the case of matrix valued H e (x). They approximately diagonalize the Hamiltonian H ε on the full space H, while we first reduce to an appropriate adiabatic subspace corresponding to the electronic levels of interest and then approximate the Hamiltonian on that subspace.…”