ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Sanin A. L., Smirnovsky A. A. Driven oscillations of quantum wave packets in system with friction, quadratic potential and impenetrable walls. Izvestiya VUZ. Applied Nonlinear Dynamics, 2007, vol. 15, iss. 4, pp. 68-83. DOI: 10.18500/0869-6632-2007-15-4-68-83

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 162)
Article type: 

Driven oscillations of quantum wave packets in system with friction, quadratic potential and impenetrable walls

Sanin Andrej Leonardovich, Peter the Great St. Petersburg Polytechnic University
Smirnovsky Aleksandr Andreevich, Peter the Great St. Petersburg Polytechnic University

The quantum dissipative system with quadratic potential confined by infinite walls of well and subjected to impulse pump was investigated in detail. The numerical simulation was carried out in context of the Schrodinger-Langevin-Kostin equation. The propagation of quantum wave packets, calculations of phase trajectories and mappings, dynamical averages, frequency spectra have been performed and discussed. These data allow to state the existence of the stable oscillatory regimes and correspondence with classic analogous systems.  

Key words: 
  1. Ford GW, Kac M, Mazur P. Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 1965;6(4):504–515. DOI: 10.1063/1.1704304.
  2. Kostin MD. On the Schrodinger-Langevin equation. J. Chem. Phys. 1972;57(9):3589–3591.
  3. Wagner HJ. Schrodinger quantization and variational principles of dissipative quantum theory. Z. Phys. B. 1994;95:261–273.
  4. Albrecht K. A new class of Schrodinger operators for quantized friction. Phys. Lett. B. 1975;56(2):127–129. DOI: 10.1016/0370-2693(75)90283-X.
  5. Hasse RW. On the quantum mechanical treatment of dissipative systems. J. Math. Phys. 1975;16(10):2005–2011. DOI: 10.1063/1.522431.
  6. Doebner HD, Goldin GA. Introducing nonlinear gauge transformation in a family of nonlinear Schrodinger equations. Phys. Rev. A. 1996;54(5):3764–3771. DOI: 10.1103/PhysRevA.54.3764.  
  7. Wysocki RJ. Hydrodynamic quantization of mechanical systems. Phys. Rev. A. 2005;72(3):032113–1. DOI: 10.1103/PhysRevA.72.032113.
  8. Van P, Fulop T. Stability of stationary solutions of the Schrodinger-Langevin equation. Phys. Lett. A. 2004;323:374–381. DOI: 10.1016/j.physleta.2004.02.035.
  9. Sanin AL. Quantum electron transport in a space with a homogeneous positive charge and a light wave. Optics and Spectroscopy. 1994;77(5):822–826. (in Russian).
  10. Andronov AA, Witt AA, Haikin SE. Theory of oscillations. Moscow: Fizmatlit; 1959. (in Russian).
  11. Landa PS. Self-oscillations in systems with finite number of degrees of freedom. Moscow : Nauka; 1980. (in Russian).
  12. Rabinovich MI, Trubetskov DI. Introduction to the Theory of Oscillations and Waves. Moscow: Nauka; 1984. (in Russian).
  13. Migulin VV, Medvedev VI, Mustel' ER, Parygin VN. Foundations of the Theory of Oscillations. Moscow: Nauka; 1988. (in Russian).
  14. Landa PS. Nonlinear Oscillations and Waves. Moscow: Fizmatlit; 1997. (in Russian).
  15. Karlov NV, Kirichenko NA. Vibrations, waves, structures. Moscow, Fizmatlit; 2001. (in Russian).
  16. Arrowsmith D, Place C. Ordinary differential equations : a qualitative approach with applications. Moscow: Mir; 1986. (in Russian).
  17. Grindlay J. On an application of a generalization of the discrete Fourier transform to short time series. Can. J. Phys. 2001;79:857–868. DOI: 10.1139/p01-054.
  18. Mott N, Sneddon I. Wave mechanics and its applications. Oxford and the Clarendon Press; 1948. 427 p.
  19. Bagmanov AT, Sanin AL. Quantum dynamics of microparticles in one-dimensional systems. Scientific and Technical Bulletin of SPbSPU. 2005;4:7–17. (in Russian).
  20. Bagmanov AT, Sanin AL. Resonances of Spatially Limited Quantum Oscillator. Telecommunications and Radio Engineering. 2005;12:46–54. (in Russian).
  21. Ushveridze AG. Dissipative quantum mechanics. A special Doebner-Goldin equation, its properties and exact solutions. Phys. Lett. A. 1994;185:123–127.
  22. Ushveridze AG. The special Doebner-Goldin equation as a fundamental equation of dissipative quantum mechanics. Phys. Lett. A. 1994;185:128–132.
  23. Smirnovsky AA, Sanin AL. Temporal resonances and structures in quantum systems under dissipation. Proceedings of SPAS jointly with UWM. Tenth Intern. Workshop on NDTSC-2006. 5–8 July 2006. Univ. of Warmia and Mazury in Olsztyn. Olsztyn, Poland. 2006;10:43–47.
Short text (in English):
(downloads: 56)