On the averaged quantum dynamics by white-noise Hamiltonians with and without dissipation

Annalen der Physik (Leipzig)

Volumn 7, Issue 2, 1998, Pages 59-100

  Fischer, Werner ^a,c   Leschke, Hajo ^a   Müller, Peter ^a,b  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

^b UNIVERSITY OF GÖTTINGEN   (Germany)

^c SIEMENS AG   (Germany)

Author keywords

Dissipation; Random quantum dynamical systems; White noise

Indexed keywords

MATHEMATICAL MODELS; RANDOM PROCESSES; WHITE NOISE;

WEYL-WIGNER SYMBOLS; WHITE NOISE HAMILTONIANS;

QUANTUM THEORY;

EID: 0031632332     PISSN: 00033804     EISSN: None     Source Type: Journal    
DOI: 10.1002/andp.2090070203     Document Type: Article

References (132)

1. G.S. Agarwal, E. Wolf, Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics I-III. Phys. Rev. D 2 (19970) 2161-2186, 2187-2205, 2206-2225
2. R. Alicki, K. Lendi, Quantum-dynamical semigroups and applications. Springer, Berlin 1987
3. P.W. Anderson, Absence of diffusion in certain random lattices. Phys. Rev. 109 (1958) 1492-1505
4. J. Anglin, S. Habib, Classical dynamics for linear systems: The case of quantum Brownian motion. Mod. Phys. Lett. A 11 (1996) 2655-2662
5. A. Arai, On a model of a harmonic oscillator coupled to a quantized, massless, scalar quantum field. I, II. J. Math. Phys. 22 (1981) 2539-2548, 2549-2552
6. T. Arai, Some extensions of semiclassical limit ℏ → 0 for Wigner functions on phase space. J. Math. Phys. 36 (1995) 622-630
7. N.L. Balasz, B.K. Jennings, Wigner's function and other distribution functions on Mock phase spaces, Phys. Rep. 104 (1984) 347-391
8. T. Banks, L. Susskind, M.E. Peskin, Difficulties for the evolution of pure states into mixed states. Nucl. Phys. B 244 (1984) 125-134
9. H. Bauer, Probability theory. Walter de Gruyter, Berlin 1996. German original: 4th ed., Walter de Gruyter, Berlin 1991
10. R. Bausch, Bewegungsgesetze nicht abgeschlossener Quantensysteme. Z. Phys. 193 (1966) 246-265 (in German)
11. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quantization. I, II. Ann. Phys. (N.Y.) 111 (1978) 61-110, 111-151
12. J.S. Bell, Speakable and unspeakable in quantum mechanics. Cambridge University Press, Cambridge 1987
13. R. Benguria, M. Kac, Quantum Langevin equation. Phys. Rev. Lett. 46 (1981) 1-4
14. V.L. Bonch-Bruevich, R. Enderlein, B. Esser, R. Keiper, A.G. Mironov, I.P. Zvyagin, Elektronentheorie ungeordneter Halbleiter. VEB Deutscher Verlag der Wissenschaften, Berlin 1984 (in German) Russian original: Nauka, Moscow 1981
15. J.P. Bouchaud, D. Touati, D. Sornette, Waves in a rapidly varying random potential: a numerical study. Phys. Rev. Lett. 68 (1992) 1787-1790
16. Z. Brzeźniak, F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations. Stochastic Proc. Appl. 55 (1995) 329-358
17. A.O. Caldeira, A.J. Leggett, Path integral approach to quantum Brownian motion. Physica A 121 (1983) 587-616. Erratum: ibid. 130 (1985) 374
18. R. Carmona, J. Lacroix, Spectral theory of random Schrödinger operators. Birkhäuser, Boston 1990
19. S.E. Cheremshantsev, Averaging of the solution of the Schrödinger equation with a randomly time-dependent potential. Sov. Phys. Dokl. 27 (1982) 718-720. Russian original: Dokl. Akad. Nauk. SSSR 266 (1982) 597-601
20. S.E. Cheremshantsev, Averaging of the solution of the Schrödinger equation with a randomly time-dependent potential. Sov. Phys. Dokl. 27 (1982) 718-720. Russian original: Dokl. Akad. Nauk. SSSR 266 (1982) 597-601
21. P. Chvosta, Exact solution of a stochastic dimer problem with single-site energy modulation. Physica A 178 (1991) 168-194
22. L. Cohen, Generalized phase-space distributions. J. Math. Phys. 7 (1966) 781-786
23. I. Daubechies, On the distributions corresponding to bounded operators in the Weyl correspondence. Commun. Math. Phys. 75 (1980) 229-238
24. I. Daubechies, Continuity statements and counterintuitive examples in connection with Weyl quantization. J. Math. Phys. 24 (1983) 1453-1461
25. E.B. Davies, The harmonic oscillator in a heat bath. Commun. Math. Phys. 33 (1973) 171-186
26. G. Doetsch, Handbuch der Laplace-Transformation, vol. 1. Birkhäuser, Basel 1950 (in German)
27. M.D. Donsker, On function space integrals. In: W.T. Martin, I. Segal (eds.). Analysis in function space, The M.I.T. Press, Cambridge (Mass.) 1964, pp. 17-30
28. J.L. Doob, Stochastic processes. Wiley, New York 1964
29. G.V. Efimov, W. von Waldenfels, On the instability of an oscillator in a field. Ann. Phys. (N.Y.) 233 (1994) 182-213
30. A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Physik (4. Folge) 17 (1905) 549-560 (in German)
31. B. Fedosov, Deformation quantization and index theory. Akademie Verlag, Berlin 1996
32. D.K. Ferry, H.L. Grubin, Modeling of quantum transport in semiconductor devices. In: H. Ehrenreich, F. Spaepen (eds.). Solid State Physics, vol. 49. Academic, San Diego 1995, pp. 283-448
33. R.P. Feynman, F.L. Vernon, Jr., The theory of a general quantum system interacting with a linear dissipative system. Ann. Phys. (N.Y.) 24 (1963) 118-173
34. H. Fink, H. Leschke, Phase-space- and other representations of quantum mechanics: a unifying approach. In: H.D. Doebner, W. Scherer, F. Schroeck, Jr. (eds.). Classical and quantum systems - foundations and symmetries. World Scientific, Singapore 1993, pp. 416-419
35. W. Fischer, H. Leschke, P. Müller, Dynamics by white-noise Hamiltonians. Phys. Rev. Lett. 73 (1994) 1578-1581
36. G.B. Folland, Harmonic analysis in phase space. Princeton University Press, Princeton 1989
37. G.W. Ford, M. Kac, P. Mazur, Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 6 (1965) 504-515
38. G.W. Ford, M.Kac, On the quantum Langevin equation. J. Stat. Phys. 46 (1987) 803-810
39. G.W. Ford, J.T. Lewis, R.F. O'Connell, Quantum Langevin equation, Phys. Rev. A 37 (1988a) 4419-4428
40. G.W. Ford, J.T. Lewis, R.F. O'Connell, Independent oscillator model of a heat bath: exact diagonalization of the Hamiltonian. J. Stat. Phys. 53 (1988b) 439-455
41. G.W. Ford, R.F. O'Connell, Derivative of the hyperbolic cotangent. Nature (Lond) 380 (1996) 113-114
42. K. Furutsu, On the statistical theory of electromagnetic waves in a fluctuating medium (1). J. Res. Nat. Bur. Stand. 67D (1963) 303-323
43. M. Gadella, L.M. Nieto, Fermion systems and the Moyal representation of quantum mechanics. J. Phys. A 26 (1993) 6043-6053
44. M. Gadella, L.M. Nieto, On the Moyal representation of quantum identical particles. Fortschr. Phys. 42 (1994) 261-279
45. C.W. Gardiner, Quantum noise. Springer, Berlin 1991
46. G.C. Ghirardi, A. Rimini, Old and new ideas in the theory of quantum measurement. In: A.I. Miller (ed.). Sixty-two years of uncertainty. Plenum, New York 1990, pp. 167-191
47. G.C. Ghirardi, A. Rimini, T. Weber, Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34 (1986) 470-491
48. G.C. Ghirardi, A. Rimini, T. Weber, Disentanglement of quantum wave functions: answer to "Comment on 'Unified dynamics for microscopic and macroscopic systems'". Phys. Rev. D 36 (1987) 3287-3289
49. S.M. Girvin, G.D. Mahan, Exact results for the quantum diffusion of many particles in a stochastic medium. Phys. Rev. B 20 (1979) 4896-4901
50. J. Glimm, A. Jaffe, Quantum physics: a functional integral point of view, 2nd ed. Springer, New York 1987
51. L. Golubović, S. Feng, F.-A. Zeng, Classical and quantum superdiffusion in a time-dependent random potential. Phys. Rev. Lett. 67 (1991) 2115-2118
52. V. Gorini, A. Kossakowski, E.C.G. Sudarshan, Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17 (1976) 821-825
53. H. Grabert, P. Schramm, G.-L. Ingold, Quantum Brownian motion: the functional integral approach, Phys. Rep. 168 (1988) 115-207
54. F. Haake, R. Reibold, Strong damping and low-temperature anomalies for the harmonic oscillator. Phys. Rev. A 32 (1985) 2462-2475
55. S. Habib, Quantum diffusion. preprint hep-th/9410181, 1994
56. P. Hänggi, P. Jung, Colored noise in dynamical systems. Adv. Chem. Phys. 89 (1995) 239-326
57. H. Haken, P. Reineker, The coupled coherent and incoherent motion of excitons and its influence on the line shape of optical absorption. Z. Phys. 249 (1972) 253-268
58. H. Haug, A.-P. Jauho, Quantum kinetics in transport and optics of semiconductors. Springer, Berlin 1996
59. S.W. Hawking, Breakdown of predictability in gravitational collapse. Phys. Rev. D 14 (1976) 2460-2473
60. S.W. Hawking, The unpredictability of quantum gravity, Commun. Math. Phys. 87 (1982) 395-414
61. S.W. Hawking, Non-trivial topologies in quantum gravity. Nucl. Phys. B 244 (1984) 135-146
62. J. Heinrichs, Quantum treatment of Brownian motion and influence of dissipation on diffusion in dynamically disordered systems. Z. Phys. B 50 (1983a) 269-273
63. J. Heinrichs, Continuum tight binding models and quantum transport in the presence of dynamical disorder. Z. Phys. B 53 (1983b) 175-181
64. J. Heinrichs, Diffusion and superdiffusion of a quantum particle in time-dependent random potentials. Z. Phys. B 89 (1992) 115-121. Erratum: ibid. 100 (1996) 327-328
65. J. Heinrichs, Quantum transport in a time-dependent random potential with a finite correlation time. Z. Phys. B 102 (1996) 137-144
66. M. Hillary, R.F. O'Connell, M.O. Scully, E.P. Wigner, Distribution functions in physics: fundamentals. Phys. Rep. 106 (1984) 121-167
67. W. Horsthemke, R. Lefever, Noise-induced transitions, theory and applications in Physics, Chemistry, and Biology. Springer, Berlin 1984
68. Y. Inaba, Diffusion of a quantum particle in a fluctuating environment. J. Phys. Soc. Japan 50 (1981) 2473-2480
69. V. Jakšić, C.-A. Pillet, Spectral theory of thermal relaxation. J. Math. Phys. 38 (1997a) 1757-1780
70. V. Jakšić, C.-A. Pillet, Ergodic properties of the non-Markovian Langevin equation. Lett. Math. Phys. 41 (1997b) 49-57
71. A.P. Jauho, A model of dynamical disorder in a uniform time-dependent electric field. J. Phys. A 20 (1987) 2895-2902
72. A.M. Jayannavar, Wave propagation and quantum superdiffusion in a rapidly varying random potential. Phys. Rev. E 48 (1993) 837-842
73. A.M. Jayannavar, N. Kumar, Nondiffusive quantum transport in a dynamically disordered medium. Phys. Rev. Lett. 48 (1982) 553-556
74. E. Joos, Comment on "Unified dynamics for microscopic and macroscopic systems." Phys. Rev. D 36 (1987) 3285-3286
75. E. Joos, H.D. Zeh, The emergence of classical properties through interaction with the environment. Z. Phys. B 59 (1985) 223-243
76. G. Jumarie, Relative information. Springer, Berlin 1990
77. I. Karatzas, S.E. Shreve, Brownian motion and stochastic calculus. Springer, Berlin 1988
78. P. Kasperkovitz, M. Peev, Wigner-Weyl formalism for toroidal geometries. Ann. Phys. (N.Y.) 230 (1994) 21-51
79. K. Kitahara, J.W. Haus, Memory effects on the quantum diffusion of a particle in a fluctuating medium. Z. Phys. B 32 (1979) 419-423
80. R. Kubo, Generalized cumulant expansion method. J. Phys. Soc. Japan 17 (1962) 1100-1120
81. R. Kubo, Stochastic Liouville equation. J. Math. Phys. 4 (1963) 174-183
82. R. Kubo, M. Toda, N. Hashitsume, Statistical physics II: nonequilibrium statistical mechanics, 2nd ed. Springer, Berlin 1991. Japanese original: Iwanami Shoten, Tokyo 1978
83. W. Kundt, Classical statistics as a limiting case of quantum statistics. Z. Naturforschg. 22a (1967) 1333-1336
84. A. Lasota, M.C. Mackey, Chaos, fractals, and noise: stochastic aspects of dynamics. 2nd ed. Springer, New York 1994
85. N. Lebedev, P. Maass, S. Feng, Diffusion and superdiffusion of a particle in a random potential with finite correlation time. Phys. Rev. Lett. 74 (1995) 1895-1899
86. E.H. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys. 31 (1990) 594-599
87. I.M. Litshits, S.A. Gredeskul, L.A. Pastur, Introduction to the theory of disordered systems. Wiley, New York 1988. Russian original: Nauka, Moscow 1982
88. G. Lindblad, On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48 (1976) 119-130
89. H. Maassen, Return to thermal equilibrium by the solution of a quantum Langevin equation. J. Stat. Phys. 34 (1984) 239-261
90. A. Madhukar, W. Post, Exact solution for the diffusion of a particle in a medium with site diagonal and off-diagonal dynamic disorder. Phys. Rev. Lett. 39 (1977) 1424-1427
91. V.B. Magalinskiǐ, Dynamical model in the theory of the Brownian motion. Sov. Phys.-JETP 9 (1959) 1381-1382. Russian original: Zh. Eksp. Teor. Fiz. 36 (1959) 1942-1944
92. V.B. Magalinskiǐ, Dynamical model in the theory of the Brownian motion. Sov. Phys.-JETP 9 (1959) 1381-1382. Russian original: Zh. Eksp. Teor. Fiz. 36 (1959) 1942-1944
93. P.A. Markowich, C.A. Ringhofer, C. Schmeiser, Semiconductor equations. Springer, Wien 1990
94. F.H. Molzahn, T.A. Osborn, A phase space fluctuation method for quantum dynamics. Ann. Phys. (N.Y.) 230 (1994) 343-394
95. J.E. Moyal, Quantum mechanics as a statistical theory. Proc. Cambridge Philos. Soc. 45 (1949) 99-124
96. F.J. Narcowich, R.F. O'Connell, Necessary and sufficient conditions for a phase-space function to be a Wigner distribution. Phys. Rev. A 34 (1986) 1-6
97. P. Neu, R. Speicher, Spectra of Hamiltonians with generalized single-site dynamical disorder. Z. Phys. B 95 (1994) 101-111
98. E.A. Novikov, Functionals and the random-force method in turbulence theory. Sov. Phys.-JETP 20 (1965) 1290-1294. Russian original: Zh. Eksp. Teor. Fiz. 47 (1964) 1919-1926
99. E.A. Novikov, Functionals and the random-force method in turbulence theory. Sov. Phys.-JETP 20 (1965) 1290-1294. Russian original: Zh. Eksp. Teor. Fiz. 47 (1964) 1919-1926
100. T.A. Osborn, F.H. Molzahn, Moyal quantum mechanics: the semiclassical Heisenberg dynamics. Ann. Phys. (N.Y.) 231 (1995) 79-127
101. A.A. Ovchinnikov, N.S. Érikhman, Motion of a quantum particle in a stochastic medium. Sov. Phys.-JETP 40 (1975) 733-737. Russian original: Zh. Eksp. Teor. Fiz. 67 (1974) 1474-1483
102. A.A. Ovchinnikov, N.S. Érikhman, Motion of a quantum particle in a stochastic medium. Sov. Phys.-JETP 40 (1975) 733-737. Russian original: Zh. Eksp. Teor. Fiz. 67 (1974) 1474-1483
103. L. Pastur, A. Figotin, Spectra of random and almost-periodic operators. Springer, Berlin 1992
104. P. Pearle, E. Squires, Bound state excitation, nucleon decay experiments, and models of wave function collapse. Phys. Rev. Lett. 73 (1994) 1-5
105. C.-A. Pillet, Some results on the quantum dynamics of a particle in a Markovian potential. Commun. Math. Phys. 102 (1985) 237-254
106. C.-A. Pillet, Asymptotic completeness for a quantum particle in a Markovian short-range potential. Commun. Math. Phys. 105 (1986) 259-280
107. J.C.T. Pool, Mathematical aspects of the Weyl correspondence. J. Math. Phys. 7 (1966) 66-76
108. M. Reed, B. Simon, Methods of modern mathematical physics III: scattering theory. Academic, New York 1979
109. S.L. Robinson, Semiclassical mechanics for time-dependent Wigner functions. J. Math. Phys. 34 (1993) 2185-2205
110. M.N. Rosenbluth, Comment on "Classical and quantum superdiffusion in a time-dependent random potential". Phys. Rev. Lett. 69 (1992) 1831
111. S.M. Rytov, Yu.A. Kravtsov, V.I. Tatarskiǐ, Principles of statistical radiophysics, vol. 3. Springer, Berlin 1989. Russian original: 2nd ed. Nauka, Moscow 1978
112. L. Saul, M. Kardar, N. Read, Directed waves in random media. Phys. Rev. A 45 (1992) 8859-8866
113. Yu.M. Shirokov, Quantum and classical mechanics in the phase space representation. Sov. J. Part. Nucl. 10 (1979) 1-18. Russian original: Fiz. Elem. Chastits At. Yadra 10 (1979) 5-50
114. Yu.M. Shirokov, Quantum and classical mechanics in the phase space representation. Sov. J. Part. Nucl. 10 (1979) 1-18. Russian original: Fiz. Elem. Chastits At. Yadra 10 (1979) 5-50
115. B.I. Shklovskii, A.L. Efros, Electronic properties of doped semiconductors. Springer, Berlin 1984. Russian original: Nauka, Moscow 1979
116. K. Sogo, Y. Fujimoto, Quantum distribution function of a nonequilibrium system. Physica A 168 (1990) 820-832
117. H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52 (1980) 569-615
118. S. Stintzing, Quantenmechanik eines Teilchens in einem zeitabhängigen Zufallspotential. Diploma Thesis, Ludwig-Maximilians-Universität München 1992 (in German)
119. W. Sutherland, A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin, Phil. Mag. (6th Series) 9 (1905) 781-785
120. V.I. Tatarskiǐ, The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26 (1983) 311-327. Russian original: Usp. Fiz. Nauk. 139 (1983) 587-619
121. V.I. Tatarskiǐ, The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26 (1983) 311-327. Russian original: Usp. Fiz. Nauk. 139 (1983) 587-619
122. S. Tcheremchantsev, Markovian Anderson model: bounds for the rate of propagation. Commun. Math. Phys. 187 (1997) 441-469
123. W. Thirring, A course in mathematical physics 4: quantum mechanics of large systems. Springer, Wien 1983. German original: Springer, Wien 1980
124. P. Ullersma, An exactly solvable model for Brownian motion I-IV. Physica 32 (1966) 27-55, 56-73, 74-89, 90-96
125. U. Weiss, Quantum dissipative systems. World Scientific, Singapore 1993
126. R.F. Werner, The classical limit of quantum theory. Preprint quant-ph/9504016. Submitted to J. Math. Phys.
127. H. Weyl, Quantenmechanik und Gruppentheorie. Z. Phys. 46 (1928) 1-46
128. E.P. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40 (1932) 749-759
129. E.P. Wigner, Quantum-mechanical distribution functions revisited. In: W. Yourgrau, A. van der Merwe (eds.). Perspectives in quantum theory. The M.I.T. Press, Cambridge (Mass.) 1971, pp. 25-36
130. W.H. Zurek, Decoherence and the transition from quantum to classical. Physics Today, October 1991, 36-44. Commented by J.L. Anderson; G.C. Ghirardi, R. Grassi, P. Pearle; N. Gisin; D.Z. Albert, G. Feinberg; P.R. Holland; V. Ambegaokar; K.J. Epstein, Physics Today, April 1993, 13-84. Reply to Comments Physics Today, April 1993, 84-90
131. W.H. Zurek, Decoherence and the transition from quantum to classical. Physics Today, October 1991, 36-44. Commented by J.L. Anderson; G.C. Ghirardi, R. Grassi, P. Pearle; N. Gisin; D.Z. Albert, G. Feinberg; P.R. Holland; V. Ambegaokar; K.J. Epstein, Physics Today, April 1993, 13-84. Reply to Comments Physics Today, April 1993, 84-90
132. W.H. Zurek, Decoherence and the transition from quantum to classical. Physics Today, October 1991, 36-44. Commented by J.L. Anderson; G.C. Ghirardi, R. Grassi, P. Pearle; N. Gisin; D.Z. Albert, G. Feinberg; P.R. Holland; V. Ambegaokar; K.J. Epstein, Physics Today, April 1993, 13-84. Reply to Comments Physics Today, April 1993, 84-90