[PDF]    https://doi.org/10.3952/physics.v57i4.3597

Open access article / Atviros prieigos straipsnis

Lith. J. Phys. 57, 195–200 (2017)


BOUND STATE INEQUALITY FROM THE SPINLESS SALPETER EQUATION WITH THE YUKAWA POTENTIAL
Jiao-Kai Chen
School of Physics and Information Science, Shanxi Normal University, 041004 Linfen, China
chenjk@sxnu.edu.cn; chenjkphy@yahoo.com

Received 13 August 2017; revised 24 September 2017; accepted 20 December 2017

In this paper, we discuss the bound-state problem for the spinless Salpeter equation with the Yukawa potential. Due to the nonlocal term of the Hamiltonian encountered, we use the eigenfunction for the ground state of the hydrogen atom as a trial function and employ the variational method to solve the spinless Salpeter equation. We derive the upper bounds on the eigenvalues to obtain the bound state inequality. The constraint on the interaction strength α is given, (2.42m–1.32μ)μ/(2.37m2) ≤ α < 8/(3π). And the maximum of the screening parameter of the Yukawa potential μ is obtained, μmax = 1.14 m.
Keywords: relativistic bound states, Yukawa potential, spinless Salpeter equation
PACS: 03.65.Ge, 03.65.Pm, 12.39.Pn, 11.10.St

SURIŠTOSIOS BŪSENOS NELYGYBĖ REMIANTIS SOLPITERIO LYGTIMI SU JUKAVOS POTENCIALU
 Jiao-Kai Chen

Šansi universiteto Fizikos ir informacijos mokslų mokykla, Linfenas, Kinija


References / Nuorodos

[1] H. Yukawa, On the interaction of elementary particles. I., Proc. Phys. Math. Soc. Jpn. 17, 48–57 (1935),
https://doi.org/10.11429/ppmsj1919.17.0_48
[2] P. Debye and E. Hückel, Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen, Phys. Z. 24, 185–206 (1923)
[3] N.W. Ashcroft and N.D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976)
[4] Y.C. Lin, C.Y. Lin, and Y.K. Ho, Spectral data of helium atoms with screened Coulomb potentials using the B-spline approach, Phys. Rev. A 85, 042516 (2012),
https://doi.org/10.1103/PhysRevA.85.042516
[5] P. Serra and S. Kais, Ground-state stability and criticality of two-electron atoms with screened Coulomb potentials using the B-splines basis set, J. Phys. B 45, 235003 (2012),
https://doi.org/10.1088/0953-4075/45/23/235003
[6] S.L. Garavelli and F.A. Oliveira, Analytical solution for a Yukawa-type potential, Phys. Rev. Lett. 66, 1310–1313 (1991),
https://doi.org/10.1103/PhysRevLett.66.1310
[7] J.P. Edwards, U. Gerber, C. Schubert, M.A. Trejo, and A. Weber, The Yukawa potential: ground state energy and critical screening, PTEP 2017(8), 083A01 (2017),
https://doi.org/10.1093/ptep/ptx107
[8] W. Lucha and F.F. Schöberl, The spinless relativistic Yukawa problem, Int. J. Mod. Phys. A 29(31), 1450195 (2014),
https://doi.org/10.1142/S0217751X14501954
[9] S. De Leo and P. Rotelli, Amplification of coupling for Yukawa potentials, Phys. Rev. D 69, 034006 (2004),
https://doi.org/10.1103/PhysRevD.69.034006
[10] F. Gross, Relativistic Quantum Mechanics and Field Theory (John Wiley & Sons, New York, 1993),
https://doi.org/10.1002/9783527617333
[11] W. Greiner, Relativistic Quantum Mechanics: Wave Equations (Springer-Verlag, New York, 2000),
https://doi.org/10.1007/978-3-662-04275-5
[12] A. Arda and R. Sever, Effective-mass Klein–Gordon–Yukawa problem for bound and scattering states, J. Math. Phys. 52, 092101 (2011),
https://doi.org/10.1063/1.3641246
[13] E.Z. Liverts and V.B. Mandelzweig, Analytical computation of amplification of coupling in relativistic equations with Yukawa potential, Ann. Phys. 324, 388–407 (2009),
https://doi.org/10.1016/j.aop.2008.08.004
[14] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators (Academic Press, New York, 1978),
https://www.elsevier.com/books/iv-analysis-of-operators/reed/978-0-08-057045-7
[15] A. Weinstein and W. Stenger, Methods of Intermediate Problems for Eigenvalues – Theory and Ramifications (Academic Press, New York, 1972),
https://www.elsevier.com/books/methods-of-intermediate-problems-for-eigenvalues-theory-and-ramifications/weinstein/978-0-12-742450-7
[16] G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators,
https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/
[17] C.T.H. Baker, The Numerical Treatment of Integral Equations (Oxford University Press, New York, 1977)
[18] L.M. Delves and J.L. Mohamed, Computational Methods for Integral Equations (Cambridge University Press, New York, 1985),
https://doi.org/10.1017/CBO9780511569609
[19] D.P. Stanley and D. Robson, Nonperturbative potential model for light and heavy quark–antiquark systems, Phys. Rev. D 21, 3180–3196 (1980),
https://doi.org/10.1103/PhysRevD.21.3180
[20] W. Lucha and F.F. Schöberl, Bound states by the spinless Salpeter equation, Fizika B 8, 193–206 (1999),
[PDF]
[21] A. Gara, B. Durand, L. Durand, and L.J. Nickisch, Relativistic description of quark–antiquark bound states. 1. Spin independent treatment, Phys. Rev. D 40, 843–854 (1989),
https://doi.org/10.1103/PhysRevD.40.843
[22] J.K. Chen, Generalized virial theorem and its application to the Salpeter equation, Acta Phys. Pol. B 47, 1155–1163 (2016),
https://doi.org/10.5506/APhysPolB.47.1155
[23] J.K. Chen, Numerical solutions of the quadratic form of the spinless Salpeter-type equation, Rom. J. Phys. 62, 119 (2017),
[PDF]
[24] E.E. Salpeter and H.A. Bethe, A relativistic equation for bound state problems, Phys. Rev. 84, 1232–1242 (1951),
https://doi.org/10.1103/PhysRev.84.1232
[25] E.E. Salpeter, Mass corrections to the fine structure of hydrogen-like atoms, Phys. Rev. 87, 328–342 (1952),
https://doi.org/10.1103/PhysRev.87.328
[26] I.W. Herbst, Spectral theory of the operator (p2 + m2)1/2Ze2/r, Commun. Math. Phys. 53, 285–294 (1977),
https://doi.org/10.1007/BF01609852
[27] A. Martin and S.M. Roy, Semi-relativistic stability and critical mass of a system of spinless bosons in gravitational interaction, Phys. Lett. B 233, 407–411 (1989),
https://doi.org/10.1016/0370-2693(89)91331-2