ASYMPTOTIC SOLUTION OF THE DIRAC EQUATION

Journal Article · Sun Sep 15 04:00:00 EDT 1963 · Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D

DOI:https://doi.org/10.1103/PhysRev.131.2789· OSTI ID:4637360

Rubinow, S I; Keller, J B

The WKB method is applied to solve the Dirac equation and the modified Dirac equation appropriate to a spin-1/2 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. (auth)

• Research Organization: Stevens Inst. of Tech., Hoboken, N.J.
• NSA Number: NSA-17-036854
• OSTI ID: 4637360
• Journal Information: Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D, Journal Name: Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D Vol. Vol: 131; ISSN PHRVA
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

BINDING ENERGY
BOUND STATE
CONFIGURATION
DIFFERENTIAL EQUATIONS
DIRAC EQUATIONS
ELECTRIC POTENTIAL
ELECTROMAGNETIC FIELDS
FERMIONS
FREQUENCY
HAMILTON-JACOBI EQUATIONS
HAMILTONIAN
MAGNETIC MOMENTS
MATHEMATICS
MECHANICS
MOTION
PHYSICS
POLARIZATION
PROBABILITY
QUANTUM MECHANICS
RELATIVITY THEORY
SPIN
SPINOR
STATISTICS
VECTORS
WKB APPROXIMATION