najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2015-6-2-224-243najo-927Research ArticlePAPERS, PRESENTED AT THE CONFERENCEPAPERS, PRESENTED AT THE CONFERENCETime-dependent quantum circular billiardBabajanovD. B.

17. Niyazov Str., 100095, Tashkent

d_babajanov@yahoo.comMatrasulovD. U.

17. Niyazov Str., 100095, Tashkent

SobirovZ. A.

60A, Amir Temur Str., 100000 Tashkent

AvazbaevS. K.

ARC Centre for Antimatter-Matter Studies, Department of Applied Physics

G.P.O. Box U1987, Perth 6845

KarpovaO. V.

17. Niyazov Str., 100095, Tashkent

National University of Uzbekistan; Turin Polytechnic University in TashkentUzbekistanNational University of Uzbekistan; Tashkent Financial InstituteUzbekistanCurtin UniversityAustralia20151408202562224243Copyright © Babajanov D.B., Matrasulov D.U., Sobirov Z.A., Avazbaev S.K., Karpova O.V., 20252025Babajanov D.B., Matrasulov D.U., Sobirov Z.A., Avazbaev S.K., Karpova O.V.Babajanov D.B., Matrasulov D.U., Sobirov Z.A., Avazbaev S.K., Karpova O.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/927

The motion of a quantum particle in a time-dependent circular billiard is studied on the basis of the Schr¨odinger equation with time-dependent boundary conditions. The cases of monotonically expanding (contracting), non-harmonically, harmonically breathing circles the case when billiard wall suddenly disappears are explored in detail. The exact analytical solutions for monotonically expanding and contracting circles are obtained. For all cases, the time-dependence of the quantum average energy is calculated. It is found that for an harmonically breathing circle, the average energy is time-periodic in the adiabatic regime with the same period as that of the oscillation. For intermediate frequencies which are comparable with the initial frequency of the particle in unperturbed billiard, such periodicity is broken. However, for very high frequencies, the average energy once again becomes periodic. A qualitative analysis of the border between adiabatic and non-adiabatic regimes is provided.

quantum billiardsSchr¨odinger equationThis work is supported in part by a grant of Volkswagen Foundation (Nr. I/82 136).ReferencesSt¨ockmann H.-J. Quantum Chaos: An Introduction. Cambridge University Press, Cambridge, UK (1999).St¨ockmann H.-J. Quantum Chaos: An Introduction. Cambridge University Press, Cambridge, UK (1999).Eckhardt B. Quantum mechanics of classically non-integrable systems. Phys. Rep., 163, P. 205–297 (1988).Eckhardt B. Quantum mechanics of classically non-integrable systems. Phys. Rep., 163, P. 205–297 (1988).Gutzwiller M.C. Chaos in Classical and Quantum Mechanics. Springer, New York (1990).Gutzwiller M.C. Chaos in Classical and Quantum Mechanics. Springer, New York (1990).Loskutov A.Yu., Ryabov A.B. and Akinshin L.G. Properties of some chaotic billiards with time-dependent boundaries. J. Phys. A, 33, P. 7973 (2000).Loskutov A.Yu., Ryabov A.B. and Akinshin L.G. Properties of some chaotic billiards with time-dependent boundaries. J. Phys. A, 33, P. 7973 (2000).Loskutov A.Yu. and Ryabov A.B. Particle Dynamics in Time-Dependent Stadium-Like Billiards. J. Stat. Phys., 108, P. 995–1014 (2002).Loskutov A.Yu. and Ryabov A.B. Particle Dynamics in Time-Dependent Stadium-Like Billiards. J. Stat. Phys., 108, P. 995–1014 (2002).De Carvalho R.E., De Souza F.C. and Leonel E.D. Fermi acceleration on the annular billiard: a simplified version, J. Phys. A: Math. Gen., 39, P. 3561(2006).De Carvalho R.E., De Souza F.C. and Leonel E.D. Fermi acceleration on the annular billiard: a simplified version, J. Phys. A: Math. Gen., 39, P. 3561(2006).Lenz F., Diakonos F.K. and Schmelcher P. Classical dynamics of the time-dependent elliptical billiard. Phys. Rev. E, 76, P. 066213 (2007).Lenz F., Diakonos F.K. and Schmelcher P. Classical dynamics of the time-dependent elliptical billiard. Phys. Rev. E, 76, P. 066213 (2007).Lenz F., Diakonos F.K. and Schmelcher P. Tunable Fermi Acceleration in the Driven Elliptical Billiard. Phys. Rev. Lett., 100, P. 014103 (2008).Lenz F., Diakonos F.K. and Schmelcher P. Tunable Fermi Acceleration in the Driven Elliptical Billiard. Phys. Rev. Lett., 100, P. 014103 (2008).Cohen D., Wisniacki D.A. Stadium billiard with moving walls. Phys. Rev. E, 67, P. 026206 (2003).Cohen D., Wisniacki D.A. Stadium billiard with moving walls. Phys. Rev. E, 67, P. 026206 (2003).Doescher S.W. and Rice M.H. Infinite Square-Well Potential with a Moving Wall. Am. J. Phys., 37, P. 1246 (1969).Doescher S.W. and Rice M.H. Infinite Square-Well Potential with a Moving Wall. Am. J. Phys., 37, P. 1246 (1969).Munier A., Burgan J.R., Feix M. and Fijalkow E. Schr¨odinger equation with time-dependent boundary conditions. J. Math. Phys., 22, P. 1219 (1981).Munier A., Burgan J.R., Feix M. and Fijalkow E. Schr¨odinger equation with time-dependent boundary conditions. J. Math. Phys., 22, P. 1219 (1981).Pinder D.N. The contracting square quantum well. Am. J. Phys., 58, P. 54 (1990).Pinder D.N. The contracting square quantum well. Am. J. Phys., 58, P. 54 (1990).Razavy M. Time-dependent harmonic oscillator confined in a box. Phys. Rev. A, 44, P. 2384 (1991).Razavy M. Time-dependent harmonic oscillator confined in a box. Phys. Rev. A, 44, P. 2384 (1991).Pereshogin P., Pronin P. Effective Hamiltonian and Berry phase in a quantum mechanical system with time dependent boundary conditions. Phys. Lett. A, 156, P. 12–16 (1991).Pereshogin P., Pronin P. Effective Hamiltonian and Berry phase in a quantum mechanical system with time dependent boundary conditions. Phys. Lett. A, 156, P. 12–16 (1991).Scheininger C. and Kleber M. Quantum to classical correspondence for the Fermi-acceleration model. Physica D, 50, P. 391–404 (1991).Scheininger C. and Kleber M. Quantum to classical correspondence for the Fermi-acceleration model. Physica D, 50, P. 391–404 (1991).Makowski A.J. and Dembinski S.T. Exactly solvable models with time-dependent boundary conditions. Phys. Lett. A, 154, P. 217–220 (1991).Makowski A.J. and Dembinski S.T. Exactly solvable models with time-dependent boundary conditions. Phys. Lett. A, 154, P. 217–220 (1991).Makowski A.J. and Peplowski P. On the behaviour of quantum systems with time-dependent boundary conditions. Phys. Lett. A, 163, P. 143–151 (1992).Makowski A.J. and Peplowski P. On the behaviour of quantum systems with time-dependent boundary conditions. Phys. Lett. A, 163, P. 143–151 (1992).Makowski A.J. Two classes of exactly solvable quantum models with moving boundaries. J. Phys. A: Math. Gen., 25, P. 3419 (1992).Makowski A.J. Two classes of exactly solvable quantum models with moving boundaries. J. Phys. A: Math. Gen., 25, P. 3419 (1992).Willemsen J.E. Exact solution of the wave dynamics of a particle bouncing chaotically on a periodically oscillating wall. Phys. Rev. E, 50, P. 3116 (1994).Willemsen J.E. Exact solution of the wave dynamics of a particle bouncing chaotically on a periodically oscillating wall. Phys. Rev. E, 50, P. 3116 (1994).Morales D.A., Parra Z., Almeida R. On the solution of the Schr¨odinger equation with time dependent boundary conditions. Phys. Lett. A, 185, P. 273–276 (1994).Morales D.A., Parra Z., Almeida R. On the solution of the Schr¨odinger equation with time dependent boundary conditions. Phys. Lett. A, 185, P. 273–276 (1994).C. Yuce. Singular potentials and moving boundaries in 3D. Phys. Lett. A, 321, P. 291–294 (2004).C. Yuce. Singular potentials and moving boundaries in 3D. Phys. Lett. A, 321, P. 291–294 (2004).Jose J.V., Gordery R. Study of a quantum fermi-acceleration model. Phys. Rev. Lett., 56, P. 290 (1986).Jose J.V., Gordery R. Study of a quantum fermi-acceleration model. Phys. Rev. Lett., 56, P. 290 (1986).Karner G. On the quantum Fermi accelerator and its relevance to ’quantum chaos’. Lett. Math. Phys. A, 17, P. 329–339 (1989).Karner G. On the quantum Fermi accelerator and its relevance to ’quantum chaos’. Lett. Math. Phys. A, 17, P. 329–339 (1989).Seba P. Quantum chaos in the Fermi-accelerator model. Phys. Rev. A, 41, P. 2306 (1990).Seba P. Quantum chaos in the Fermi-accelerator model. Phys. Rev. A, 41, P. 2306 (1990).Jana T.K., Roy P. A class of exactly solvable Schr¨odinger equation with moving boundary condition. Phys. Lett. A, 372, P. 2368–2373 (2008).Jana T.K., Roy P. A class of exactly solvable Schr¨odinger equation with moving boundary condition. Phys. Lett. A, 372, P. 2368–2373 (2008).www.netlib.orgwww.netlib.orgGodoy S. Diffraction in time: Fraunhofer and Fresnel dispersion by a slit. Phys. Rev. A, 65, P. 042111 (2002).Godoy S. Diffraction in time: Fraunhofer and Fresnel dispersion by a slit. Phys. Rev. A, 65, P. 042111 (2002).Glasser M.L., Mateo J., Negro J. and Nieto L.M. Quantum infinite square well with an oscillating wall. Chaos, Solitons and Fractals, 41, P. 2067–2074 (2009).Glasser M.L., Mateo J., Negro J. and Nieto L.M. Quantum infinite square well with an oscillating wall. Chaos, Solitons and Fractals, 41, P. 2067–2074 (2009).Koiller J., Markarian R., Kamphorst S.O. and De Carvalho S.P. Time-dependent billiards. Nonlinearity, 8, P. 983 (1995).Koiller J., Markarian R., Kamphorst S.O. and De Carvalho S.P. Time-dependent billiards. Nonlinearity, 8, P. 983 (1995).Kamphorst S.O., De Carvalho S.P. Bounded gain of energy on the breathing circle billiard. Nonlinearity, 12, P. 1363-1371 (1999).Kamphorst S.O., De Carvalho S.P. Bounded gain of energy on the breathing circle billiard. Nonlinearity, 12, P. 1363-1371 (1999).Badrinarayanan R., Jose J.V., Chu G. Quantum manifestations of classical chaos in a Fermi accelerating disk. Physica D, 83, P. 1–29 (1995).Badrinarayanan R., Jose J.V., Chu G. Quantum manifestations of classical chaos in a Fermi accelerating disk. Physica D, 83, P. 1–29 (1995).Burgio G.F., Baldo M., Rapisarda A., Schuck P. Chaoticity in vibrating nuclear billiards. Phys. Rev. C, 52, P. 2475 (1995).Burgio G.F., Baldo M., Rapisarda A., Schuck P. Chaoticity in vibrating nuclear billiards. Phys. Rev. C, 52, P. 2475 (1995).Robinett R.W., Heppelmann S. Quantum wave-packet revivals in circular billiards. Phys. Rev. A, 65, P. 062103 (2002).Robinett R.W., Heppelmann S. Quantum wave-packet revivals in circular billiards. Phys. Rev. A, 65, P. 062103 (2002).Robinett R.W. Quantum mechanics of the two-dimensional circular billiard plus baffle system and half-integral angular momentum. Eur. J. Phys., 24, P. 231 (2003).Robinett R.W. Quantum mechanics of the two-dimensional circular billiard plus baffle system and half-integral angular momentum. Eur. J. Phys., 24, P. 231 (2003).Abramowitz M., Stegun I.A. Handbook of mathematical functions. M.: Nauka. (1964).Abramowitz M., Stegun I.A. Handbook of mathematical functions. M.: Nauka. (1964).

The authors declare that there are no conflicts of interest present.