Asymptotic properties of quantum dynamics in bounded domains at various time scales

I. V. Volovich ; A. S. Trushechkin

Izvestiya. Mathematics , Tome 76 (2012) no. 1, pp. 39-78.

Résumé

We study a peculiar semiclassical limit of the dynamics of quantum states on a circle and in a box (infinitely deep potential well with rigid walls) as the Planck constant tends to zero and time tends to infinity. Our results describe the dynamics of coherent states on the circle and in the box at all time scales in semiclassical approximation. They give detailed information about all stages of quantum evolution in the semiclassical limit. In particular, we rigorously justify the fact that the spatial distribution of a wave packet is most often close to a uniform distribution. This fact was previously known only from numerical experiments. We apply the results obtained to a problem of classical mechanics: deciding whether the recently suggested functional formulation of classical mechanics is preferable to the traditional one. To do this, we study the semiclassical limit of Husimi functions of quantum states. Both formulations of classical mechanics are shown to adequately describe the system when time is not arbitrarily large. But the functional formulation remains valid at larger time scales than the traditional one and, therefore, is preferable from this point of view. We show that, although quantum dynamics in finite volume is commonly believed to be almost periodic, the probability distribution of the position of a quantum particle in a box has a limit distribution at infinite time if we take into account the inaccuracy in measuring the size of the box.

Détail

Keywords: dynamics of quantum systems, semiclassical limit, weak limit.

Bibliographie

[1] P. Bocchieri, A. Loinger, “Quantum recurrence theorem”, Phys. Rev. (2), 107:2 (1957), 337–338 | DOI | MR | Zbl

[2] I. Sh. Averbukh, N. F. Perelman, “Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics”, Phys. Rev. Lett., 139:9 (1989), 449–453 | DOI

[3] I. Sh. Averbukh, N. F. Perel'man, “The dynamics of wave packets of highly-excited states of atoms and molecules”, Soviet Phys. Uspekhi, 34:7 (1991), 572–591 | DOI

[4] D. L. Aronstein, C. R. Stroud, “Fractional wave-function revivals in the infinite square well”, Phys. Rev. A, 55:6 (1997), 4526–4537 | DOI

[5] R. W. Robinett, “Visualizing the collapse and revival of wave packets in the infinite square well using expectation values”, Amer. J. Phys., 68:5 (2000), 410–420 | DOI

[6] R. W. Robinett, “Quantum wave packet revivals”, Phys. Rep., 392:1–2 (2004), 1–119 | DOI | MR

[7] E. M. Wright, D. F. Walls, J. C. Garrison, “Collapses and revivals of Bose–Einstein condensates formed in small atomic samples”, Phys. Rev. Lett., 77:11 (1996), 2158–2161 | DOI

[8] P. Plötz, J. Madroñero, S. Wimberger, “Collapse and revival in inter-band oscillations of a two-band Bose–Hubbard model”, J. Phys. B, 43:8 (2010) | DOI

[9] I. V. Volovich, A. S. Trushechkin, “Squeezed quantum states on an interval and uncertainty relations for nanoscale systems”, Proc. Steklov Inst. Math., 265 (2009), 276–306 | DOI | MR | Zbl

[10] A. Puankare, “Zamechaniya o kineticheskoi teorii gazov”, Izbrannye trudy v trekh tomakh, v. III, Matematika. Teoreticheskaya fizika. Analiz matematicheskikh i estestvennonauchnykh rabot A. Puankare, Nauka, M., 1974, 385–412 | MR | Zbl

[11] V. V. Kozlov, Teplovoe ravnovesie po Gibbsu i Puankare, In-t kompyuternykh issledovanii, M.–Izhevsk, 2002 | MR | Zbl

[12] V. V. Kozlov, Ansambli Gibbsa i neravnovesnaya statisticheskaya mekhanika, RKhD, M.–Izhevsk, 2008

[13] L. Accardi, Yu. G. Lu, I. Volovich, Quantum theory and its stochastic limit, Springer-Verlag, Berlin, 2002 | MR | Zbl

[14] I. V. Volovich, “Problema neobratimosti i funktsionalnaya formulirovka klassicheskoi mekhaniki”, Vestn. SamarGU, 8:67 (2008), 35–55

[15] I. V. Volovich, “Randomness in classical mechanics and Quantum mechanics”, Found. Phys., 41:3 (2011), 516–528 | DOI | Zbl

[16] I. V. Volovich, “Bogoliubov equations and functional mechanics”, Theoret. and Math. Phys., 164:3 (2010), 1128–1135 | DOI

[17] A. S. Trushechkin, I. V. Volovich, “Functional classical mechanics and rational numbers”, P-Adic Numbers Ultrametric Anal. Appl., 1:4 (2009), 361–367 | DOI | MR

[18] A. S. Trushechkin, “Irreversibility and the role of an instrument in the functional formulation of classical mechanics”, Theoret. and Math. Phys., 164:3 (2010), 1198–1201 | DOI

[19] V. V. Kozlov, D. V. Treshchev, “Fine-grained and coarse-grained entropy in problems of statistical mechanics”, Theoret. and Math. Phys., 151:1 (2007), 539–555 | DOI | MR | Zbl

[20] V. V. Kozlov, O. G. Smolyanov, “Wigner function and diffusion in a collision-free medium of quantum particles”, Theory Probab. Appl., 51:1 (2007), 168–181 | DOI | MR | Zbl

[21] J. R. Klauder, E. C. G. Sudarshan, Fundamentals of quantum optics, Benjamin, New York–Amsterdam, 1968 | MR

[22] J. R. Klauder, B.-S. Skagerstam, Coherent states. Applications in physics and mathematical physics, World Scientific, Singapore, 1985 | MR | Zbl

[23] G. A. González, M. A. del Olmo, “Coherent states on the circle”, J. Phys. A, 31:44 (1998), 8841–8857 | DOI | MR | Zbl

[24] A. A. Karatsuba, S. M. Voronin, The Riemann zeta-function, de Gruyter Exp. Math., 5, de Gruyter, Berlin, 1992 | MR | MR | Zbl | Zbl

[25] D. Mumford, Tata lectures on theta, Birkhaüser, Boston, MA, 1983 | MR | MR | Zbl

[26] V. S. Vladimirov, Equations of mathematical physics, Dekker, New York, 1971 | MR | MR | Zbl | Zbl

[27] V. P. Maslov, M. V. Fedoryuk, Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR | Zbl

[28] K. Inoue, M. Ohya, I. V. Volovich, “Semiclassical properties and chaos degree for the quantum baker's map”, J. Math. Phys., 43 (2002), 734–755 | DOI | MR | Zbl

[29] L. D. Landau, E. M. Lifshitz, Course of theoretical physics, v. 3, Quantum mechanics: non-relativistic theory, Pergamon Press, Oxford, 1958 | MR | MR | Zbl | Zbl

[30] E. V. Piskovskiy, I. V. Volovich, “On the correspondence between Newtonian and functional mechanics”, Quantum Bio-Informatics IV. From Quantum Infarmation to Bio-Informatics, QP-PQ: Quantum Probab. White Noise Anal., 28, World Scientific, Hackensack, NJ, 2011, 363–372

[31] K. Husimi, “Some formal properties of the density matrix”, Proc. Phys.-Math. Soc. Japan, III. Ser., 22 (1940), 264–314 | Zbl

[32] J. McKenna, H. L. Frisch, “Quantum-mechanical, microscopic Brownian motion”, Phys. Rev. (2), 145:1 (1966), 93–110 | DOI | MR

[33] M. Hillery, R. F. O'Connel, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals”, Phys. Rep., 106:3 (1984), 121–167 | DOI | MR

[34] L. D. Faddeev, O. A. Yakubovskiǐ, Lectures on quantum mechanics for mathematics students, Stud. Math. Libr., 47, Amer. Math. Soc., Providence, RI, 2009 | MR | Zbl | Zbl

[35] V. V. Kozlov, D. V. Treshchev, “Weak convergence of solutions of the Liouville equation for nonlinear hamiltonian systems”, Theoret. and Math. Phys., 134:3 (2003), 339–350 | DOI | MR | Zbl