Abstract:
We outline a recursive method for obtaining WKB expansions of solutions of the Dirac equation in an external centrally symmetric field with a scalar-vector Lorentz structure of the interaction potentials. We obtain semiclassical formulas for radial functions in the classically allowed and forbidden regions and find conditions for matching them in passing through the turning points. We generalize the Bohr–Sommerfeld quantization rule to the relativistic case where a spin-1/2 particle interacts simultaneously with a scalar and an electrostatic external field. We obtain a general expression in the semiclassical approximation for the width of quasistationary levels, which was earlier known only for barrier-type electrostatic potentials (the Gamow formula). We show that the obtained quantization rule exactly produces the energy spectrum for Coulomb- and oscillatory-type potentials. We use an example of the funnel potential to demonstrate that the proposed version of the WKB method not only extends the possibilities for studying the spectrum of energies and wave functions analytically but also ensures an appropriate accuracy of calculations even for states with nr∽1.
Citation:
V. Yu. Lazur, A. K. Reity, V. V. Rubish, “WKB method for the Dirac equation with a scalar-vector coupling”, TMF, 143:1 (2005), 83–111; Theoret. and Math. Phys., 143:1 (2005), 559–582
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\paper WKB method for the Dirac equation with a scalar-vector coupling
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\jour Theoret. and Math. Phys.
\yr 2005
\vol 143
\issue 1
\pages 559--582
\crossref{https://doi.org/10.1007/s11232-005-0090-1}
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Linking options:
https://www.mathnet.ru/eng/tmf1805
https://doi.org/10.4213/tmf1805
https://www.mathnet.ru/eng/tmf/v143/i1/p83
This publication is cited in the following 18 articles:
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Lazur V.Yu., Reity O.K., Rubish V.V., “Quasiclassical theory of the Dirac equation with a scalar-vector interaction and its applications in the physics of heavy-light mesons”, Phys. Rev. D, 83:7 (2011), 076003, 23 pp.
Sun S., Ang L.K., Shiffler D., Luginsland J.W., “Klein tunnelling model of low energy electron field emission from single-layer graphene sheet”, Applied Physics Letters, 99:1 (2011), 013112
Lazur V.Yu., Reity O.K., Rubish V.V., “Spherical model of the Stark effect in external scalar and vector fields”, Int. J. Mod. Phys. A, 25:16 (2010), 3235–3259
Alhaidari A.D., “Relativistic Coulomb Problem for Z Larger Than 137”, Internat J Modern Phys A, 25:18–19 (2010), 3703–3714
Esposito, G, “On the phase-integral method for the radial Dirac equation”, Journal of Physics A-Mathematical and Theoretical, 42:39 (2009), 395203
V. Yu. Lazur, A. K. Reity, V. V. Rubish, “Semiclassical approximation in the relativistic potential model of
B and D mesons”, Theoret. and Math. Phys., 155:3 (2008), 825–847
Van Orden, JW, “Scaling of Dirac fermions and the WKB approximation”, Physical Review D, 72:5 (2005), 054020
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