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On the Quantization of One-Dimensional Conservative Systems with Variable mass

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The Hamiltonian associated to the mass variable system is constructed from first principles through finding a constant of motion of the system. A comparison is made of the classical motion of a body with its mass position depending in the (

*x,v*) space and (*x,p*) space which are defined by the constant of motion and the Hamiltonian, for a particular model of mass variation. As one could expected, these motion looks different on these spaces. The quantization of the harmonic oscillator with this mass variation is done, and a comparison is made by using the usual Hamiltonian approach with the proposed quantization of the constant of motion approach. This comparison is done at first order in perturbation theory, and one sees a difference between both approaches which can, in principle, be measured.Conflicts of Interest

The authors declare no conflicts of interest.

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G. López, "On the Quantization of One-Dimensional Conservative Systems with Variable mass,"

*Journal of Modern Physics*, Vol. 3 No. 8, 2012, pp. 777-785. doi: 10.4236/jmp.2012.38102.

[1] | H. Goldstein, “Classical Mechanics,” Addison-Wesley, Boston, 1950. |

[2] | F. W. Helhl, C. Kiefer and R. J. K. Metzler, “Black Holes: Theory and Observation,” Springer-Verlag, Berlin, 1998. |

[3] | P. W. Daly, “The Use of Kepler Trajectories to Calculate Ion Fluxes at Multi-Gigameter Distances from Comet Halley,” Astronomy and Astrophysics, Vol. 226, No. 1, 1989, pp. 318-334. |

[4] | H. Gylden, “Die Bahnbewegungen in einem Systeme von zwei K?rpern in dem Falle, dass die Massen Ver?nderungen unterworfen sind,” Astronomy and Astrophysics, Vol. 109, No. 1-2, 1884, pp. 1-6. doi:10.1002/asna.18841090102 |

[5] | I. V. Meshcherskii, “Ein Specialfall des Gyldn’schen Problems,” Astronomische Nachrichten, Vol. 132, No. 3153, 1893, p. 93. |

[6] | I. V. Meshcherskii, “Ueber die Integration der Bewegung- sgleichungen im Problemezweier Krper von vernderlicher Masse,” Astronomische Nachrichten, Vol. 159, No. 15, 1902, pp. 229-242. |

[7] | E. O. Lovett, “Note on Gyldén’s Equations of the Prob- lem of Two Bodies with Masses Varying with the Time,” Astronomische Nachrichten, Vol. 158, No. 2, 1902, pp. 337-344. doi:10.1002/asna.19021582202 |

[8] | J. H. Jeans, “The Effect of Varying Mass on a Binary System,” Monthly Notices of the Royal Astronomical Society, Vol. 85, 1925, p. 912. |

[9] | L. M. Berkovich, “Gylden-Me??erskii Problem,” Celes- tial Mechanics and Dynamical Astronomy, Vol. 24, No. 4, 1981, pp. 407-429. doi:10.1007/BF01230399 |

[10] | A. A. Bekov, “Integrable Cases and Motion Trajectories in the Gylden-Meshcherskii Problem,” Soviet Astronomy, Vol. 33, 1989, pp. 71-78. |

[11] | C. Prieto and J. A. Docobo, “Analythic Solution of the Two-Body Problem with Slowly Decreasing Mass,” Astronomy and Astrophysics, Vol. 318, 1997, pp. 657-661. |

[12] | G. V. López, “About Galilean Transformation on a Mass Variable System and Two Bodies Gravitational System with Variable Mass and Dampen-Anti Damping Effect Due to Star Wind,” 2012. http://arxiv.org/abs/1203.0495v1 |

[13] | H. A. Bethe, “Possible Explanation of the Solar-Neutrino Puzzle,” Physical Review Letters, Vol. 56, No. 12, 1986, pp. 1305-1308. doi:10.1103/PhysRevLett.56.1305 |

[14] | E. D. Commins and P. H. Bucksbaum, “Weak Interactions of Leptons and Quarks,” Cambridge University Press, Cambridge, 1983. |

[15] | A. G. Zagorodny, P. P. J. M. Schram and S. A. Trigger, “Stationary Velocity and Charge Distributions of Grains in Dusty Plasmas,” Physical Review Letters, Vol. 84, No. 16, 2000, pp. 3594-3597. doi:10.1103/PhysRevLett.84.3594 |

[16] | O. T. Serimaa, J. Javanainen and S. Varró, “Gauge-Inde- pendent Wigner functions: General Formulation,” Physi- cal Review A, Vol. 33, No. 5, 1986, pp. 2913-2927. doi:10.1103/PhysRevA.33.2913 |

[17] | I. Ye. Terapov, “On Some Fundamental Problems of the Variable-Mass Continuum Mechanics,” International Journal of Fluid Mechanics Research, Vol. 28, No. 4, 2001, pp. 152-174. |

[18] | C. Quesne, B. Bagchi, A. Banerjee and V. M. Tkachuk, Hamiltonians with Position-Dependent Mass, Deforma- tions and Supersymmetry,” Bulgarian Journal of Physics, Vol. 33, 2006, pp. 308-318. |

[19] | Y. Hamdouni, “Motion of Position-Dependent Effective Mass as a Damping-Antidamping Process: Application to the Fermi Gas and the Morse Potential,” Journal of Phys- ics A: Mathematical and Theoretical, Vol. 44, No. 38, 2011, Article ID: 385301. doi:10.1088/1751-8113/44/38/385301 |

[20] | M. ?apak, Y. Can?elik and ?. L. ünsal, S. Tay and B. G?nül, “An Extended Scenario for the Schr?dinger Equa- tion,” Journal of Mathematical Physics, Vol. 52, No. 10, 2011, Article ID: 102102. doi:10.1063/1.3646371 |

[21] | J. A. Kobussen, “Some Comments on the Lagrangian Formalism for Systems with General Velocity Dependent Forces,” Acta Physica Austriaca, Vol. 51, 1979, pp. 293- 309. |

[22] | C. Leubner, “Inequivalent Lagrangians from Constants of the Motion,” Physical Review A, Vol. 86, No. 2, 1981, pp. 68-70. doi:10.1016/0375-9601(81)90166-3 |

[23] | G. López, “One-Dimensional Autonomous Systems and Dissipative Systems,” Annals of Physics, Vol. 251, No. 2, 1996, pp. 372-383. doi:10.1006/aphy.1996.0118 |

[24] | G. Lópezand, and G. González, “Quantum Bouncer with Dissipation,” International Journal of Theoretical Phys- ics, Vol. 43, No. 10, 2004, pp. 1999-2008. doi:10.1023/B:IJTP.0000049005.73750.c0 |

[25] | G. López and P. López, “Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscilla- tor,” International Journal of Theoretical Physics, Vol. 45, No. 4, 2006, pp. 753-742. doi:10.1007/s10773-006-9064-9 |

[26] | G. López, “Restricted Constant of Motionfor the One- Dimensional Harmonic Oscillator with Quadratic Dissi- pation and Some Consequences in Statistic and Quantum Mechanics,” International Journal of Theoretical Physics, Vol. 79, No. 4, 2001, pp. 71-79. doi:10.1023/A:1011972700121 |

[27] | A. Messiah, “Quantum Mechanics Vol. I,” John Wiley and Sons, New York, 1958. |

[28] | P. A. M. Dirac, “The Principles of Quantum Mechanics,” 4th Edition, Oxford Science Publications, Oxford, 1992. |

[29] | H. Weyl, “Quantenmechanik und Gruppentheorie,” Zeits- chrift für Physick, Vol. 46, No. 1-2, 1927, pp. 1-46. doi:10.1007/BF02055756 |

[30] | R. Kubo, “Wigner Representation of Quantum Operators and Its Applications to Electrons in a Magnetic Field,” Journal of the Physical Society of Japan, Vol. 19, 1964, pp. 2127-2139. doi:10.1143/JPSJ.19.2127 |

[31] | C. Cohen-Tannoudji, B. Diu and F. Lalo?, “Quantum Mechanics Vol. I,” John Wiley and Sons, New York, 1977. |

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