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

Physical Review

Asymptotic Solution of the Dirac Equation

Sol I. Rubinow, Joseph Bishop Keller

Published Sep 15, 1963• 130 citations
Publisher

Abstract

The WKB method is applied to solve the Dirac equation and the modified Dirac equation appropriate to a spin-\textonehalf{} particle with an anomalous magnetic moment. The solution consists of a phase factor multiplied by a spinor amplitude which is a power series in Planck's constant. The phase is a solution of the Hamilton-Jacobi equation of relativistic mechanics for a spinless particle without electric or magnetic moments. Each term in the spinor amplitude satisfies an ordinary differential equation along the relativistic trajectories. The equation for the leading amplitude yields an equation for the polarization four-vector which is identical with that derived classically by Bargmann, Michel, and Telegdi. It also yields the law of conservation of probability in a tube of trajectories. In addition, it gives rise to an equation for a supplementary phase factor. By using the classical Hamilton-Jacobi function, the law of probability conservation, the polarization four-vector and the supplementary phase factor, the leading term in the solution of the Dirac or modified Dirac equation can be constructed. This solution should be useful when the wavelength of the particle is small compared to the characteristic distance associated with the electromagnetic potential through which the particle moves. When applied to the bound states of a particle without an anomalous moment in a spherically symmetric electrostatic potential, it yields the same results as are usually obtained by separation of variables and use of the ordinary WKB method. The advantage of the present method is that it applies equally well to nonseparable problems.

DOI
10.1103/physrev.131.2789

Related papers

The complex WKB-Maslov method for the Dirac equation in a torsion field: I. Construction of trajectory-coherent states and the equation for spin

Vladislav G. Bagrov, Владимир Владимирович Белов, Andrey Yu. Trifonov +1 more

The investigation of the Dirac equation by the WKB-Maslov method (complex sprout method) in Minkowski-Cartan space with an arbitrary external electromagnetic field is carried out. The Ehrenfest-type complete set of quasi-classical trajectory-coherent states (TCS) for the Dirac equation up to the order of O(h(cross) 3 2/) at to 0 is constructed. These states have the form of wavepackets moving along the worldline of the charged particle. From the Dirac equation a classical relativistic spin equation is derived by the use of the TCS which coincides with the generalization of the well known Bargmann-Michel-Telegdi equation in the case of an arbitrary external torsion field.

Classical and Quantum Gravity•Jul 1, 1991•10 citations

КВАНТОВАЯ МЕХАНИКА ЭЛЕКТРОНА В МАГНИТНОМ ПОЛЕ, УЧЕТ АНОМАЛЬНОГО МАГНИТНОГО МОМЕНТА

Е. М. Овсиюк, О. В. Веко, Я. А. Войнова +2 more

The Dirac equation for spin 1/2 particle with anomalous magnetic moment is solved in presence of the external uniform magnetic field. After separation of the variables, the problem is reduced to a 4-order ordinary differential equation, which is solved exactly with the use of the factorization method. A generalized formulas for Landau energy levels are found. Solutions are expressed in terms of confluent hypergeometric functions.

Doklady of the National Academy of Sciences of Belarus•Oct 29, 2016•0 citations

Classical Motions of Spin-½ Particles

Kenneth Rafanelli, Ralph Schiller

From the WKB solutions to the squared Dirac equation we derive the classical trajectories and spin precessions originally postulated by Bargmann, Michel, and Telegdi for charged spin-\textonehalf{} particles. Identical equations of motion are obtained from the WKB solutions to the Dirac equation.

Physical Review•Jul 13, 1964•54 citations

The hydrodynamic form of the Dirac equation and the distorted wave Glauber approximation

F. Riordan

The Dirac equation for an electron subjected to a potential A mu is shown to be equivalent to the following classical equations for a beam of non-interacting electrons subjected to A mu in addition to 'quantum' scalar and four-vector potentials together with an anomalous electromagnetic field affecting the spin: the Hamilton-Jacobi equation for the momentum, the continuity equation for the charge distribution, Frenkel's relativistic equation for the spin and a new equation giving a phase theta as an integral along the electron's path. These equations are coupled through the 'quantum' potentials and fields. In the classical limit theta is constant along the electron's path and the equations decouple. Two approximation procedures based on these coupled hydrodynamic equations are given: (1) the semiclassical to lowest order in powers of h(cross), (2) the distorted wave Glauber approximation assumes the solution of the Dirac equation and hence of the hydrodynamic equations for an unperturbed A mu and that the electrons travel along the same four-dimensional paths in the perturbed potential, but with different 'speed'.

Journal of Physics A Mathematical and General•Jan 11, 1983•2 citations

The Dirac Equation Derived from the Relativistic Coulomb Field: Stochastic Mechanics in Four-Dimensional Spacetime

Fusao Ishii

This paper extends the stochastic electrodynamic framework of Papers 1, 2, and 3 [1–3] to relativistic quantum mechanics, deriving the Dirac equation from the relativistic Coulomb field. The non-relativistic Brownian motion of Paper 1 is generalised to a four-dimensional stochastic process in Minkowski spacetime, parameterised by proper time τ . The relativistic diffusion coefficient isD4D = ℏ/m—exactly twice the non-relativistic value ℏ/2m—arising from the additional contribution of the temporal component of the four-dimensional Wiener process. The covariant zero-point field spectral tensor Sμν(ω) = (ℏω^3/6π^2ϵ0c^3)gμν is derived from the relativistic Coulomb field through the Lorentz-covariant Einstein–Hopf detailed balance and the ergodic theorem, whose applicability to the SED electron-vacuum system is proved in Paper 3 [3]. Relativistic stochastic optimal control gives the Hamilton–Jacobi–Bellman (HJB) equation in four-dimensional configuration space, which yields the Klein–Gordon equation for spin-0 particles through the relativistic Itô correction (ℏ/2m)□. Linearisation of the relativistic Lagrangian −mc√uμuμ via Dirac matrices γμ satisfying {γμ, γν} = 2gμν converts the second-order HJB equation into a first-order equation—the Dirac equation: (iℏγμ∂μ−eγμAμ/c−mc)ψ = 0. The central new result is that spin- 1/2 is not introduced by the linearisation: it is already present in the geometry of the four-dimensional stochastic path. The mass-shell constraint uμuμ = −c^2 forces the relativistic Brownian path to be helical, with radius ℏ/mc and angular frequency 2mc^2/ℏ (Zitterbewegung). The angular momentum of this helix is ℏ/2 (spin- 1/2 ) and the magnetic moment gives g = 2—both following from the Coulomb field without additional assumptions. The linearisation then finds the unique first-order wave equation consistent with this pre-existing helical spin structure. The four-component Dirac spinor encodes four degrees of freedom already determined by the stochastic path: two spin states (helix handedness) and two time directions (particle/antiparticle).

•Jun 17, 2026•0 citations

Dirac equation for electrodynamic model particles

J. X. Zheng-Johansson

We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c ; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.

Journal of Physics Conference Series•Aug 1, 2008•5 citations
The WKB method is applied to solve the Dirac equation and the modified Dirac equation appropriate to a spin-\textonehalf{} particle with an anomalous magnetic moment. The solution consists of a phase factor multiplied by a spinor amplitude which is a power series in Planck's constant. The phase is a solution of the Hamilton-Jacobi equation of relativistic mechanics for a spinless particle without electric or magnetic moments. Each term in the spinor amplitude satisfies an ordinary differential equation along the relativistic trajectories. The equation for the leading amplitude yields an equation for the polarization four-vector which is identical with that derived classically by Bargmann, Michel, and Telegdi. It also yields the law of conservation of probability in a tube of trajectories. In addition, it gives rise to an equation for a supplementary phase factor. By using the classical Hamilton-Jacobi function, the law of probability conservation, the polarization four-vector and the supplementary phase factor, the leading term in the solution of the Dirac or modified Dirac equation can be constructed. This solution should be useful when the wavelength of the particle is small compared to the characteristic distance associated with the electromagnetic potential through which the particle moves. When applied to the bound states of a particle without an anomalous moment in a spherically symmetric electrostatic potential, it yields the same results as are usually obtained by separation of variables and use of the ordinary WKB method. The advantage of the present method is that it applies equally well to nonseparable problems.
The investigation of the Dirac equation by the WKB-Maslov method (complex sprout method) in Minkowski-Cartan space with an arbitrary external electromagnetic field is carried out. The Ehrenfest-type complete set of quasi-classical trajectory-coherent states (TCS) for the Dirac equation up to the order of O(h(cross) 3 2/) at to 0 is constructed. These states have the form of wavepackets moving along the worldline of the charged particle. From the Dirac equation a classical relativistic spin equation is derived by the use of the TCS which coincides with the generalization of the well known Bargmann-Michel-Telegdi equation in the case of an arbitrary external torsion field.
The Dirac equation for an electron subjected to a potential A mu is shown to be equivalent to the following classical equations for a beam of non-interacting electrons subjected to A mu in addition to 'quantum' scalar and four-vector potentials together with an anomalous electromagnetic field affecting the spin: the Hamilton-Jacobi equation for the momentum, the continuity equation for the charge distribution, Frenkel's relativistic equation for the spin and a new equation giving a phase theta as an integral along the electron's path. These equations are coupled through the 'quantum' potentials and fields. In the classical limit theta is constant along the electron's path and the equations decouple. Two approximation procedures based on these coupled hydrodynamic equations are given: (1) the semiclassical to lowest order in powers of h(cross), (2) the distorted wave Glauber approximation assumes the solution of the Dirac equation and hence of the hydrodynamic equations for an unperturbed A mu and that the electrons travel along the same four-dimensional paths in the perturbed potential, but with different 'speed'.
This paper extends the stochastic electrodynamic framework of Papers 1, 2, and 3 [1–3] to relativistic quantum mechanics, deriving the Dirac equation from the relativistic Coulomb field. The non-relativistic Brownian motion of Paper 1 is generalised to a four-dimensional stochastic process in Minkowski spacetime, parameterised by proper time τ . The relativistic diffusion coefficient isD4D = ℏ/m—exactly twice the non-relativistic value ℏ/2m—arising from the additional contribution of the temporal component of the four-dimensional Wiener process. The covariant zero-point field spectral tensor Sμν(ω) = (ℏω^3/6π^2ϵ0c^3)gμν is derived from the relativistic Coulomb field through the Lorentz-covariant Einstein–Hopf detailed balance and the ergodic theorem, whose applicability to the SED electron-vacuum system is proved in Paper 3 [3]. Relativistic stochastic optimal control gives the Hamilton–Jacobi–Bellman (HJB) equation in four-dimensional configuration space, which yields the Klein–Gordon equation for spin-0 particles through the relativistic Itô correction (ℏ/2m)□. Linearisation of the relativistic Lagrangian −mc√uμuμ via Dirac matrices γμ satisfying {γμ, γν} = 2gμν converts the second-order HJB equation into a first-order equation—the Dirac equation: (iℏγμ∂μ−eγμAμ/c−mc)ψ = 0. The central new result is that spin- 1/2 is not introduced by the linearisation: it is already present in the geometry of the four-dimensional stochastic path. The mass-shell constraint uμuμ = −c^2 forces the relativistic Brownian path to be helical, with radius ℏ/mc and angular frequency 2mc^2/ℏ (Zitterbewegung). The angular momentum of this helix is ℏ/2 (spin- 1/2 ) and the magnetic moment gives g = 2—both following from the Coulomb field without additional assumptions. The linearisation then finds the unique first-order wave equation consistent with this pre-existing helical spin structure. The four-component Dirac spinor encodes four degrees of freedom already determined by the stochastic path: two spin states (helix handedness) and two time directions (particle/antiparticle).
We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c ; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.