Abstract
We discuss an exactly solvable relativistic model of a nonrelativistic linear harmonic oscillator in the presence of a constant external force. We show that as in the nonrelativistic case, the relativistic linear oscillator in an external uniform field is unitarily equivalent to the oscillator without this field. Using two methods, we calculate transition amplitudes between energy states of the discrete spectrum of the relativistic linear oscillator under the action of a suddenly applied uniform field. We find Barut–Girardello coherent states and the Green’s function in the coordinate and momentum representations. We obtain the linear and bilinear generating functions for the Meixner–Pollaczek polynomials. We prove that the relativistic wave functions, the generators of the dynamical symmetry group, and the transition amplitudes have the correct nonrelativistic limit.
References
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Nonrelativistic Theory, Pergamon Press, New York (1973).
A. I. Baz, Ya. B. Zeldovich, and A. M. Perelomov, Scattering, Reactions and Decay in Non-relativistic Quantum Mechanics [in Russian], Nauka, Moscow (1971).
M. Moshinsky and Yu. F. Smirnov, The Harmonic Oscillator in Modern Physics (Contemporary Concepts in Physics, Vol. 9), Harwood Academic Publ., Amsterdam (1996).
V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Course of Theoretical Physics, Vol. 4: Quantum Electrodynamics, Pergamon, New York (1982).
H. Yukawa, “Structure and mass spectrum of elementary particles. II. Oscillator model,” Phys. Rev., 91, 416–417 (1953).
M. Markov, “On dynamically deformable form factors in the theory of elementary particles,” Nuovo Cimento, 3, 760–772 (1956).
R. P. Feynman, M. Kislinger, and F. Ravndal, “Current matrix elements from a relativisti quark model,” Phys. Rev. D, 3, 2706–2732 (1971).
T. De, Y. S. Kim, and M. E. Noz, “Radial effects in the symmetric quark model,” Nuovo Cimento A, 13, 1089–1101 (1973).
Y. S. Kim and M. E. Noz, “Group theory of covariant harmonic oscillators,” Am. J. Phys., 46, 480–483 (1978).
Y. S. Kim and M. E. Noz, “Relativistic harmonic oscillators and hadronic structure in the quantum-mechanics curriculum,” Am. J. Phys., 46, 484–488 (1978).
M. Moshinsky and A. Szczepaniak, “The Dirac oscillator,” J. Phys. A: Math. Gen., 22, L817–L819 (1989).
O. L. de Lange, “Shift operators for a Dirac oscillator,” J. Math. Phys., 32, 1296–1300 (1991).
Z.-F. Li, J.-J. Liu, W. Lucha, W.-G. Ma, and F. F. Schöberl, “Relativistic harmonic oscillator,” J. Math. Phys., 46, 103514, 11 pp. (2005); arXiv: hep-ph/0501268.
K. Kowalski and J. Rembieliński, “Relativistic massless harmonic oscillator,” Phys. Rev. A, 81, 012118, 6 pp. (2010); arXiv: 1002.0474.
A. D. Donkov, V. G. Kadyshevskii, M. D. Matveev, and R. M. Mir-Kassimov, “Quasipotential equation for a relativistic harmonic oscillator,” Theoret. and Math. Phys., 8, 673–681 (1971).
N. M. Atakishiyev, R. M. Mir-Kassimov, and Sh. M. Nagiyev, “Quasipotential models of a relativistic oscillator,” Theoret. and Math. Phys., 44, 592–603 (1980).
N. M. Atakishiyev, “Quasipotential wave functions of a relativistic harmonic oscillator and Pollaczek polynomials,” Theoret. and Math. Phys., 58, 166–171 (1984).
N. M. Atakishiyev, R. M. Mir-Kasimov, and Sh. M. Nagiyev, “A relativistic model of the isotropic oscillator,” Ann. Phys., 497, 25–30 (1985).
R. M. Mir-Kasimov, Sh. M. Nagiev, and E. Dzh. Kagramanov, Relyativistskiy lineynyy ostsillyator pod deystviem postoyannoy vneshney sily i bilineynaya proizvodyashchaya funktsiya dlya polinomov Pollacheka (Preprint No. 214), SKB IFAN AzSSR, Baku (1987).
E. D. Kagramanov, R. M. Mir-Kasimov, and Sh. M. Nagiyev, “The covariant linear oscillator and generalized realization of the dynamical \(\mathrm{SU}(1,1)\) symmetry algebra,” J. Math. Phys., 31, 1733–1738 (1990).
R. M. Mir-Kasimov, “\(\mathrm{SU}_q(1,1)\) and the relativistic oscillator,” J. Phys. A: Math. Gen., 24, 4283–4302 (1991).
Yu. A. Grishechkin and V. N. Kapshai, “Solution of the Logunov–Tavkhelidze equation for the three-dimensional oscillator potential in the relativistic configuration representation,” Russ. Phys. J., 61, 1645–1652 (2019).
N. M. Atakishiyev, Sh. M. Nagiyev, and K. B. Wolf, “Realization of \(Sp(2,\mathfrak{R})\) by finite-difference operators: The relativistic oscillator in an external field,” J. Group Theor. Phys., 3, 61–70 (1995).
Sh. M. Nagiyev and R. M. Mir-Kassimov, “Relativistic linear oscillator under the action of a constant,” Theoret. and Math. Phys., 208, 1265–1276 (2021).
Yu. A. Grishechkin and V. N. Kapshai, “Approximate analytic solution of the Logunov–Tavkhelidze equation for a one-dimensional oscillator potential in the relativistic configuration representation,” Theoret. and Math. Phys., 211, 826–837 (2022).
V. G. Kadyshevsky, R. M. Mir-Kasimov, and N. B. Skachkov, “Quasi-potential approach and the expansion in relativistic spherical functions,” Nuovo Cimento A, 55, 233–257 (1968).
V. G. Kadyshevskii, R. M. Mir-Kasimov, and N. B. Skachkov, “Three-dimensional formulation of the relativistic two-body problem,” Part. Nucl., 2, 635–690 (1972).
N. M. Atakishiyev and K. B. Wolf, “Generalized coherent states for a relativistic model of the linear oscillator in a homogeneous external field,” Rep. Math. Phys., 27, 305–311 (1989).
N. M. Atakishiyev, Sh. M. Nagiyev, and K. B. Wolf, “Wigner distribution functions for a relativistic linear oscillator,” Theoret. and Math. Phys., 114, 322–334 (1998).
Sh. M. Nagiyev, G. H. Guliyeva, and E. I. Jafarov, “The Wigner function of the relativistic finite-difference oscillator in an external field,” J. Phys. A: Math. Theor., 42, 454015, 10 pp. (2009).
K. Husimi, “Miscellanea in elementary quantum mechanics, II,” Progr. Theor. Phys., 9, 381–402 (1953).
E. H. Kerner, “Note on the forced and damped oscillator in quantum mechanics,” Can. J. Phys., 36, 371–377 (1958).
Sh. M. Nagiyev and A. I. Akhmedov, “Time evolution of quadratic quantum systems: Evolution operators, propagators, and invariants,” Theoret. and Math. Phys., 198, 392–411 (2019).
A. O. Barut and R. Raczka, Theory of Group Representations and Applications, World Sci., Singapore; Polish Sci. Publ. PWN, Warszawa (1986).
A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 3: Special Functions, Gordon and Breach, New York (1986).
R. W. Fuller, S. M. Harris, and E. L. Slaggie, “\(S\)-matrix solution for the forced harmonic oscillator,” Am. J. Phys., 31, 431–439 (1963).
P. Carruthers and M. M. Nieto, “Coherent states and forced quantum oscillator,” Am. J. Phys., 33, 537–544 (1965).
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York–Toronto–London (1953).
R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer, Berlin (2010).
M. M. Nieto and D. R. Truax, “Holstein–Primakoff/Bogoliubov transformations and multiboson system,” Fortschr. Phys., 45, 145–156 (1997).
A. M. Perelomov, “Coherent states for arbitrary Lie group,” Commun. Math. Phys., 26, 222–236 (1972).
A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys., 21, 41–55 (1971).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions, Gordon and Breach, New York (1986).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflicts of interest.
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 214, pp. 81–101 https://doi.org/10.4213/tmf10337.
Appendix
We prove equality (4.7). Using the Baker–Hausdorff formula [40]
On the other hand, using operator representation (4.6) for the orthonormalized states of the linear harmonic oscillator in a uniform external field, we obtain
Rights and permissions
About this article
Cite this article
Nagiyev, S.M., Mir-Kasimov, R.M. Relativistic linear oscillator under the action of a constant external force. Transition amplitudes and the Green’s function. Theor Math Phys 214, 72–88 (2023). https://doi.org/10.1134/S004057792301004X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S004057792301004X