Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

The Reduction of the Dirac Equation to a Nonrelativistic Form

R.E. Moss

Published Oct 1, 1971• 6 citations
Publisher

Abstract

The positive and negative energy solutions of the Dirac equation for a free electron are completely decoupled using the method of small components. It is shown that this separation is formally equivalent to that obtained with the Foldy-Wouthuysen transformation but that the separation is achieved much more economically with the method of small components. The method may also be used to relate operators in the relativistic and nonrelativistic representations.

DOI
10.1119/1.1976597

Related papers

Foldy–Wouthuysen-type reduction of the second-order Dirac equation

Levere C. Hostler

A nonrelativistic reduction of the second-order Dirac equation is investigated. An equation K2+[Π0; K−Π̂]=m2+Π̂2 is obtained for a ‘‘kinetic Hamiltonian’’ K such that Φ obeys a nonrelativistic Schrödinger-type equation Π0Φ=KΦ in addition to obeying the second-order Dirac equation. Non-self-adjoint terms in K are removed by a change of representation from Φ to ΦT ≡PΦ, and a closed expression P≡[ 1/2 +(K† −Π̂)(K−Π̂)/2(m)2]1/2 is obtained for P. Similar results are obtained for antiparticle states, and a type of ‘‘factorization theorem’’ for the second-order Dirac equation is obtained in which the second-order Dirac equation is replaced by a pair of uncoupled first-order equations. Since nonperturbative techniques are used throughout, the factorization theorem holds independently of the question of convergence of the Foldy–Wouthuysen series and is valid whenever the equation K2+[Π0; K−Π̂]=m2+Π̂2 admits a solution for K. The connection with the usual linear Dirac equation is investigated, and closed expressions are described for a transformation to a representation of that equation which is ‘‘even’’ in the language of Foldy and Wouthuysen. The transformation described is equivalent to two consecutive ‘‘odd’’ unitary transformations.

Journal of Mathematical Physics•Jul 1, 1989•0 citations

Separated Two-Component Equations for Spin-½ Particles Using Foldy-Wouthuysen-Type Transformations with an Application to the k. p. Method in Solids

S. Teitler

The Foldy-Wouthuysen formalism is presented in a representation-independent manner. It is shown that an algebraic transformation serves to remove the mass term from the Dirac equation for spin-\textonehalf{} particles, thereby providing decoupled two-component equations in a particular (Weyl) representation. Also it is shown that appropriate use of a constant algebraic transformation applied to a Dirac Hamiltonian in the usual (Pauli) representation leads to a nonrelativistic k. p equation used in solid-state theory for which the k-dependent spin-orbit term is absent.

Physical Review•Dec 25, 1968•0 citations

Dirac Algebra and Foldy-Wouthuysen Transform

Heinz Otto Cordes

We offer further results and a matured physical interpretation, concerning our invariant algebra P for the Dirac equation discussed in 1983 [ 1 ] and 1996 [ 2 ]. Especially, P may be fully described either with simple spectral theory of the Hamiltonian or with a decoupling modulo order — ∞ of the positive and negative energy parts of the Dirac equation, similar to the Foldy-Wouthuysen transform. There seems to be evidence indicating that only operators in P qualify as observables—i.e., can be measured with arbitrary precision—a feature comparable to the Heisenberg uncertainty relation.

CRC Press eBooks•Apr 24, 2019•1 citation

Nonlocality of Foldy-Wouthuysen and related transformations for the Dirac equation

Tomasz M. Rusin, Wlodek Zawadzki

Nonlocalities of Foldy-Wouthuysen and related transformations, which are used to separate positive- and negative-energy states in the Dirac equation, are investigated. Second moments of functional kernels generated by the transformations are calculated, and the transformed functions and their variances are computed. It is shown that all the transformed quantities are smeared in the coordinate space by an amount comparable to the Compton wavelength ${\ensuremath{\lambda}}_{c}=\ensuremath{\hbar}/mc$.

arXiv:1108.4143•Physical Review A•Dec 29, 2011•6 citations

A non-relativistic reduction of the Dirac equation in the free-particle basis

JEAN POTVIN

Uses the non-relativistic limit of the integro-differential formulation of the Dirac equation in order to find an effective Schrodinger equation which, solved in the well known non-relativistic free-particle basis, gives approximately the same results (the probability amplitudes) as the Dirac equation for non-relativistic states written in terms of the free-particle spinors. Effective Hamiltonians for the electromagnetic and pseudoscalar interactions are derived. Comparison with the results of the Foldy-Wouthuysen method (1950) for the electromagnetic case is also achieved.

Journal of Physics A Mathematical and General•May 1, 1981•1 citation

Discussion on non-relativistic approximation of Dirac equation by means of Foldy-Wouthuysen transformation

Yun‐De Li

The non-relativistic approximation of Dirac equation is discussed by means of method of Foldy-Wouthuysen transformation.The recurrence fomiulas of Hamiltonian and wave function are given out.

College Physics•Jan 1, 2013•0 citations
The positive and negative energy solutions of the Dirac equation for a free electron are completely decoupled using the method of small components. It is shown that this separation is formally equivalent to that obtained with the Foldy-Wouthuysen transformation but that the separation is achieved much more economically with the method of small components. The method may also be used to relate operators in the relativistic and nonrelativistic representations.
A nonrelativistic reduction of the second-order Dirac equation is investigated. An equation K2+[Π0; K−Π̂]=m2+Π̂2 is obtained for a ‘‘kinetic Hamiltonian’’ K such that Φ obeys a nonrelativistic Schrödinger-type equation Π0Φ=KΦ in addition to obeying the second-order Dirac equation. Non-self-adjoint terms in K are removed by a change of representation from Φ to ΦT ≡PΦ, and a closed expression P≡[ 1/2 +(K† −Π̂)(K−Π̂)/2(m)2]1/2 is obtained for P. Similar results are obtained for antiparticle states, and a type of ‘‘factorization theorem’’ for the second-order Dirac equation is obtained in which the second-order Dirac equation is replaced by a pair of uncoupled first-order equations. Since nonperturbative techniques are used throughout, the factorization theorem holds independently of the question of convergence of the Foldy–Wouthuysen series and is valid whenever the equation K2+[Π0; K−Π̂]=m2+Π̂2 admits a solution for K. The connection with the usual linear Dirac equation is investigated, and closed expressions are described for a transformation to a representation of that equation which is ‘‘even’’ in the language of Foldy and Wouthuysen. The transformation described is equivalent to two consecutive ‘‘odd’’ unitary transformations.
The Foldy-Wouthuysen formalism is presented in a representation-independent manner. It is shown that an algebraic transformation serves to remove the mass term from the Dirac equation for spin-\textonehalf{} particles, thereby providing decoupled two-component equations in a particular (Weyl) representation. Also it is shown that appropriate use of a constant algebraic transformation applied to a Dirac Hamiltonian in the usual (Pauli) representation leads to a nonrelativistic k. p equation used in solid-state theory for which the k-dependent spin-orbit term is absent.
We offer further results and a matured physical interpretation, concerning our invariant algebra P for the Dirac equation discussed in 1983 [ 1 ] and 1996 [ 2 ]. Especially, P may be fully described either with simple spectral theory of the Hamiltonian or with a decoupling modulo order — ∞ of the positive and negative energy parts of the Dirac equation, similar to the Foldy-Wouthuysen transform. There seems to be evidence indicating that only operators in P qualify as observables—i.e., can be measured with arbitrary precision—a feature comparable to the Heisenberg uncertainty relation.
Nonlocalities of Foldy-Wouthuysen and related transformations, which are used to separate positive- and negative-energy states in the Dirac equation, are investigated. Second moments of functional kernels generated by the transformations are calculated, and the transformed functions and their variances are computed. It is shown that all the transformed quantities are smeared in the coordinate space by an amount comparable to the Compton wavelength ${\ensuremath{\lambda}}_{c}=\ensuremath{\hbar}/mc$.
Uses the non-relativistic limit of the integro-differential formulation of the Dirac equation in order to find an effective Schrodinger equation which, solved in the well known non-relativistic free-particle basis, gives approximately the same results (the probability amplitudes) as the Dirac equation for non-relativistic states written in terms of the free-particle spinors. Effective Hamiltonians for the electromagnetic and pseudoscalar interactions are derived. Comparison with the results of the Foldy-Wouthuysen method (1950) for the electromagnetic case is also achieved.