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Something New about Radial Wave Functions of Fermions in the Repulsive Coulomb Field

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Abstract

An impermeable barrier at \(r = {{r}_{{{\text{cl}}}}}\) in the effective potential of the relativistic Schrödinger-type equation leads to exclusion of the range \(0 \leqslant r < {{r}_{{{\text{cl}}}}}\) from the wave function domain. Based on the duality of the Schrödinger-type equation and the Dirac equation, a similar exclusion should be made in the wave function domain of the Dirac equation. As a result, we obtain new solutions to the Dirac equation in the Coulomb repulsion field. Calculations show that depending on working parameters, at distances of fractions or units of the Compton wavelength of the fermion from \(r = {{r}_{{{\text{cl}}}}}\) new solutions almost coincide with the standard Coulomb functions of the continuous spectrum. Practically, matrix elements with new solutions will coincide to a good accuracy with standard matrix elements with the Coulomb functions of the continuous spectrum. Our consideration is methodological and helpful for discussing further development of quantum theory.

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REFERENCES

  1. V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Quantum Electrodynamics (Nauka, Moscow, 2006; Elsevier, 2012).

  2. W. Pauli, “Die Allgemeinen Prinzipien der Wellenmechanik,” in Handbuch der Physik, Ed. By H. Geiger and K. Shell (Springer, Berlin, 1933), Vol. 24, Part 1.

    Google Scholar 

  3. Ya. B. Zel’dovich and V. S. Popov, “Electronic structure of superheavy atoms,” Sov. Phys. Usp. 14, 673 (1972).

    Article  Google Scholar 

  4. M. V. Gorbatenko and V. P. Neznamov, “Quantum mechanics of stationary states of particles in a space-time of classical black holes,” Theor. Math. Phys. 205, 1492—1526 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  5. K. M. Case, “Singular potentials,” Phys. Rev. 80, 797 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  6. L. D. Landau and E. M. Lifshits, Quantum Mechanics. Nonrelativistic Theory (Fizmatlit, Moscow, 1963; Elsevier, 2013).

  7. A. M. Perelomov and V. S. Popov, “Fall to the center” in quantum mechanics”, Theor. Math. Phys. 4, 664–6777 (1970).

    Article  Google Scholar 

  8. V. P. Neznamov and I. I. Safronov, “Stationary solutions of second-order equations for point fermions in the Schwarzschild gravitational field”, J. Exp. Theor. Phys. 127, 647–658 (2018).

    Article  Google Scholar 

  9. V. P. Neznamov, I. I. Safronov, and V. E. Shemarulin, “Stationary solutions of second-order equations for fermions in Reissner–Nordström space-time,” J. Exp. Theor. Phys. 127, 684—704 (2018).

    Article  Google Scholar 

  10. V. P. Neznamov, I. I. Safronov, and V. E. Shemarulin, “Stationary solutions of the second-order equation for fermions in Kerr-Newman space-time,” J. Exp. Theor. Phys. 128, 84 (2019).

    Google Scholar 

  11. V. P. Neznamov and I. I. Safronov, “Second-order stationary solutions for fermions in an external Coulomb field,” J. Exp. Theor. Phys. 128, 672 (2019).

    Article  Google Scholar 

  12. H. Pruefer, “Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung Stetiger Funktionen,” Math. Ann. 95, 499 (1926).

    Article  MathSciNet  MATH  Google Scholar 

  13. I. Ulehla and M. Havliček, “New method for computation of discrete spectrum,” Appl. Math 25, 257 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Ulehla, M. Havliček, and J. Hořejši, “Eigenvalues of the Schrödinger operator via the Pruefer transformation,” Phys. Lett. A 82, 64 (1981).

    Article  MathSciNet  Google Scholar 

  15. I. Ulehla, “On some applications of the Prüfer transformation and its extensions,” Preprint No. RL-82-095 (Rutherford Appleton Lab. Chilton, 1982).

    Google Scholar 

  16. L. L. Foldy and S. A. Wouthuysen, “On the Dirac theory of spin 1/2 particles and its non-relativistic limit,” Phys. Rev. 78, 29 (1950);

    Article  MATH  Google Scholar 

  17. E. Eriksen, “Foldy–Wouthuysen transformation. Exact solution with generalization to the two-particle problem,” Phys. Rev. 111, 1011 (1958);

    Article  MathSciNet  MATH  Google Scholar 

  18. V. P. Neznamov and A. J. Silenko, “Foldy–Wouthuysen wave functions and conditions of transformation between Dirac and Foldy–Wouthuysen representations,” J. Math. Phys. 50, 122302 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. J. Silenko, Pengming Zhang, and Liping Zou, “Silenko, Zhang, and Zou Reply,” Phys. Rev. Lett. 122, 159302 (2019).

    Article  Google Scholar 

  20. Liping Zou, Pengming Zhang and A. J. Silenko, “Position and spin in relativistic quantum mechanics,” Phys. Rev. A 101, 032117 (2020).

    Article  MathSciNet  Google Scholar 

  21. A. J. Silenko, “General properties of the Foldy–Wouthuysen transformation and applicability of the corrected original Foldy-Wouthuysen method,” Phys. Rev. A 93, 022108 (2016).

    Article  MathSciNet  Google Scholar 

  22. V. P. Neznamov, “On the theory of interacting fields in Foldy–Wouthuysen representation,” Phys. Part. Nucl. 37, 86—103 (2006).

    Article  Google Scholar 

  23. V. P. Neznamov, “The isotopic Foldy–Wouthuysen representation and chiral symmetry,” Phys. Part. Nucl. 43, 15–35 (2012).

    Article  Google Scholar 

  24. V. P. Neznamov and V. E. Shemarulin, “Quantum electrodynamics with self-conjugated equations with spinor wave functions for fermion fields,” Int. J. Mod. Phys. A 36, 2150086 (2021).

    Article  MathSciNet  Google Scholar 

  25. L. S. Holster, “Scalar formalism for quantum electrodynamics,” J. Math. Phys. 26, 1348 (1985).

    Article  MathSciNet  Google Scholar 

  26. V. P. Neznamov, “The lack of vacuum polarization in quantum electrodynamics with spinors in fermion equations,” Int. J. Mod. Phys. A 36, 2150173 (2021).

    Article  MathSciNet  Google Scholar 

  27. A. Lasenby, S. Dolan, J. Prutchard, A. Caceres, and S. Dolan, “Bound states and decay times of fermions in a Schwarzschild black hole background,” Phys. Rev. D 72, 105014 (2005).

    Article  Google Scholar 

  28. I. G. Petrovskii, Lectures on Partial Differential Equations (Fiz.-Mat. Lit., Moscow, 1961; Dover Publications. 1992).

  29. V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1981; Mir Publishers, 1971).

  30. J. Dittrich and P. Exner, “Tunneling through a singular potential barrier,” J. Math. Phys. 26, 2000 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, “New determination of the fine structure constant from the electron g value and QED,” Phys. Rev. Lett. 97, 030802 (2006).

    Article  Google Scholar 

  32. Yu. V. Prokhorov (Chief Ed.), Encyclopedic Dictionary of Mathematics (Sov. Entsiklopediya, Moscow, 1988) [in Russian]

  33. A. F. Andreev, Singular Points of Differential Equations (Vysshaya Shkola, Minsk, 1979) [in Russian].

    Google Scholar 

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ACKNOWLEDGMENTS

The authors are grateful to A.L. Novoselov for his substantial technical assistance in preparation of the paper.

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Authors and Affiliations

  1. Russian Federal Nuclear Center—All-Russian Research Institute of Experimental Physics, 607188, Sarov, Russia

    V. P. Neznamov, I. I. Safronov & V. E. Shemarulin

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  1. V. P. Neznamov
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  2. I. I. Safronov
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  3. V. E. Shemarulin
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Correspondence to V. P. Neznamov, I. I. Safronov or V. E. Shemarulin.

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Translated by M. Potapov

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Neznamov, V.P., Safronov, I.I. & Shemarulin, V.E. Something New about Radial Wave Functions of Fermions in the Repulsive Coulomb Field. Phys. Part. Nuclei 53, 1126–1137 (2022). https://doi.org/10.1134/S1063779622060053

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