Time-evolution of quantum systems via a complex nonlinear Riccati equation. II: Dissipative systems. (English) Zbl 1380.81171
Summary: In our former contribution [Ann. Phys. 360, 44–60 (2015; Zbl 1360.81139)], we have shown the sensitivity to the choice of initial conditions in the evolution of Gaussian wave packets via the nonlinear Riccati equation. The formalism developed in the previous work is extended to effective approaches for the description of dissipative quantum systems. By means of simple examples we show the effects of the environment on the quantum uncertainties, correlation function, quantum energy contribution and tunnelling currents. We prove that the environmental parameter \(\gamma\) is strongly related with the sensitivity to the choice of initial conditions.
MSC:
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 37L50 | Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems |
| 34M25 | Formal solutions and transform techniques for ordinary differential equations in the complex domain |
Citations:
Zbl 1360.81139References:
| [1] | Cruz, H.; Schuch, D.; Castaños, O.; Rosas-Ortiz, O., Ann. Physics, 360, 44 (2015) · Zbl 1360.81139 |
| [2] | Milne, E., Phys. Rev., 35, 863 (1930) |
| [3] | Chandrasekhar, S., Rev. Modern Phys., 15, 1 (1943) · Zbl 0061.46403 |
| [4] | Weiss, U., Quantum Dissipative Systems (1993), World Scientific: World Scientific Singapore · Zbl 1137.81300 |
| [5] | Van Kampen, N. G., Stochastic Processes in Physics and Chemistry (2004), Elsevier: Elsevier Amsterdam · Zbl 0974.60020 |
| [6] | Caldeira, A. O.; Leggett, A. J., Ann. Phys., 153, 445 (1983) |
| [7] | Gorini, V.; Kossakowski, A.; Sudarshan, E. C.G., J. Math. Phys., 17, 821 (1976) · Zbl 1446.47009 |
| [8] | Lindblad, G., Comm. Math. Phys., 48, 3101 (1976) |
| [9] | Gisin, N., J. Phys. A: Math. Gen., 14, 2259 (1981) · Zbl 0474.35040 |
| [10] | Albrecht, K., Phys. Lett. B, 56, 127 (1975) |
| [11] | Hasse, R. W., J. Math. Phys., 16, 2005 (1975) |
| [12] | Schuch, D.; Chung, K.-M., Int. J. Quantum Chem., 29, 1561 (1986) |
| [13] | Schuch, D., Phys. Rev. A, 55, 935 (1997) |
| [14] | Cruz y. Cruz, S.; Rosas-Ortiz, O., SIGMA, 9, 004 (2013) |
| [15] | Bagchi, B.; Das, S.; Ghosh, S.; Poria, S., J. Phys. A, 46, Article 032001 pp. (2013) · Zbl 1273.81102 |
| [16] | Lakshmanan, M.; Chandrasekar, V. K., Eur. Phys. J. Spec. Top., 222, 665 (2013) |
| [17] | Caldirola, P., Nuovo Cimento, 183, 93 (1941) |
| [18] | Kanai, E., Progr. Theoret. Phys., 3, 440 (1948) |
| [19] | Nassar, A. B., J. Math. Phys., 27, 755 (1986) |
| [20] | Sun, C. P.; Yu, L. H., Phys. Rev. A, 51, 1845 (1995) |
| [21] | Schrödinger, E., Ann. Phys., 79, 361 (1926) · JFM 52.0965.08 |
| [22] | Dirac, P. A.M., Lectures on Quantum Mechanics (2001), Dover: Dover New York |
| [23] | Schrödinger, E., Ann. Phys. (Berlin), 79, 734 (1926) · JFM 52.0967.02 |
| [24] | Schrödinger, E., Collected Papers on Wave Mechanics (1928), Blacke & Son: Blacke & Son London · JFM 54.0963.02 |
| [25] | Doebner, H.-D.; Goldin, G. A., J. Phys. A: Math. Gen., 27, 1771 (1994) · Zbl 0842.35113 |
| [26] | Madelung, E., Die Mathematischen Hilfsmittel des Physikers (1950), Springer: Springer Berlin · JFM 48.0246.02 |
| [27] | Mrowka, B., Z. Phys., 130, 164 (1951) · Zbl 0045.45701 |
| [28] | Schuch, D., J. Phys.: Conf. Ser., 380, Article 012009 pp. (2012) |
| [29] | Kostin, M. D., J. Chem. Phys., 57, 3589 (1972) |
| [30] | Kerr, W. C.; Graham, A. J., Eur. Phys. J. B, 15, 305 (2000) |
| [31] | Schuch, D.; Kaushal, R. S., J. Phys.: Conf. Ser.. J. Phys.: Conf. Ser., Phys. Rev. A, 55, 935 (1997) |
| [32] | Dekker, H., Phys. Rep., 80, 1 (1981) |
| [33] | Hillery, M.; OConnell, R. F.; Scully, M. O.; Wigner, E. P., Phys. Rep., 106, 121 (1984) |
| [34] | Dodonov, V. V.; Dodonov, A. V., J. Russ. Laser Res., 26, 6 (2005) |
| [35] | Degasperis, A.; Fonda, L.; Ghirardi, G. C., Nuovo Cimento A, 21, 471 (1974) |
| [36] | Misra, B.; Sudarshan, E. C.G., J. Math. Phys., 18, 756 (1977) |
| [37] | Schuch, D., Phys. Lett. A, 294, 31 (2002) · Zbl 0985.81047 |
| [38] | Schuch, D., J. Phys. A: Math. Gen., 35, 8615 (2002) · Zbl 1045.81517 |
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