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TMF, 2005, Volume 143, Number 1, Pages 83–111 (Mi tmf1805)  

This article is cited in 13 scientific papers (total in 13 papers)

WKB method for the Dirac equation with a scalar-vector coupling

V. Yu. Lazur, A. K. Reity, V. V. Rubish

Uzhgorod National University

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 $n_r\backsim1$.

Keywords: Dirac equation, Lorentz structure of interaction potential, WKB method, effective potential, quantization condition, level width, potential models


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English version:
Theoretical and Mathematical Physics, 2005, 143:1, 559–582

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Received: 04.08.2004
Revised: 15.10.2004

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

Citation in format AMSBIB
\by V.~Yu.~Lazur, A.~K.~Reity, V.~V.~Rubish
\paper WKB method for the Dirac equation with a scalar-vector coupling
\jour TMF
\yr 2005
\vol 143
\issue 1
\pages 83--111
\jour Theoret. and Math. Phys.
\yr 2005
\vol 143
\issue 1
\pages 559--582

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    This publication is cited in the following articles:
    1. Van Orden, JW, “Scaling of Dirac fermions and the WKB approximation”, Physical Review D, 72:5 (2005), 054020  crossref  adsnasa  isi  scopus  scopus
    2. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. Esposito, G, “On the phase-integral method for the radial Dirac equation”, Journal of Physics A-Mathematical and Theoretical, 42:39 (2009), 395203  crossref  mathscinet  zmath  isi  scopus  scopus
    4. 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  crossref  zmath  adsnasa  isi  elib  scopus  scopus
    5. Alhaidari A.D., “Relativistic Coulomb Problem for Z Larger Than 137”, Internat J Modern Phys A, 25:18–19 (2010), 3703–3714  crossref  zmath  adsnasa  isi  scopus  scopus
    6. 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.  crossref  adsnasa  isi  elib  scopus  scopus
    7. 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  crossref  adsnasa  isi  elib  scopus  scopus
    8. B. A. Zon, A. S. Kornev, “Semiclassical approximation of the Dirac equation in a central field”, Theoret. and Math. Phys., 171:1 (2012), 478–489  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    9. Sun S., Ang L.K., “Shot Noise of Low Energy Electron Field Emission Due to Klein Tunneling”, J. Appl. Phys., 112:1 (2012), 016104  crossref  adsnasa  isi  scopus  scopus
    10. Liang Sh.-J., Sun S., Ang L.K., “Over-Barrier Side-Band Electron Emission From Graphene with a Time-Oscillating Potential”, Carbon, 61 (2013), 294–298  crossref  isi  elib  scopus  scopus
    11. Liang Sh., Ang L.K., “Electron Over-Barrier Emission Mechanism of Single Layer Graphene”, 2013 IEEE 14th International Vacuum Electronics Conference (Ivec), IEEE, 2013  adsnasa  isi
    12. Liang Sh.-J., Ang L.K., “Chiral Tunneling-Assisted Over-Barrier Electron Emission From Graphene”, IEEE Trans. Electron Devices, 61:6, SI (2014), 1764–1770  crossref  adsnasa  isi  scopus  scopus
    13. Campos A.G., Cabrera R., Rabitz H.A., Bondar D.I., “Analytic Solutions to Coherent Control of the Dirac Equation”, Phys. Rev. Lett., 119:17 (2017), 173203  crossref  isi  scopus  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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