Abstract
A new memoryless expression for the equation of motion for the reduced density matrix is derived. It is equivalent to that proposed by Tokuyama and Mori, but has a more convenient form for the application of the perturbational expansion method. The master equation derived from this form of equation in the first Born approximation is applied to two examples, the Brownian motion of a quantal oscillator and that of a spin. In both examples the master equation is rewritten into the coherent-state representation. A comparison is made with the stochastic theory of the spectral line shape given by Kubo, and it is shown that this theory of the line shape can be incorporated into the framework of the present theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Zwanzig,Physica 30:1109 (1964); inLectures in Theoretical Physics, W. E. Brittin, B. W. Downs, and J. Downs, eds. (Interscience, New York, 1961), Vol. 3, p. 106; G. G. Emch, inLectures in Theoretical Physics, 1965 Summer Institute for Theoretical Physics, W. E. Brittin, ed. (Univ. of Colorado Press, Boulder, 1966), Vol. VIIIA, p. 65; P. N. Argyres, inLectures in Theoretical Physics, 1965 Summer Institute for Theoretical Physics, W. E. Brittin, ed. (Univ. of Colorado Press, Boulder, 1966), Vol. VIIIA, p. 183.
R. Kubo, inLectures at Sitges International School of Statistical Mechanics,1974, L. Garrid, ed. (Springer Verlag, Berlin, 1975); H. Mori,Prog. Theor. Phys. 33:423 (1965).
H. Haken,Rev. Mod. Phys. 47:67 (1975), inHandbuch der Physik, Vol. XXV/2c, L. Genzel, ed. (Springer Verlag, Berlin, 1970); R. Graham, inSpringer Tracts in Modern Physics, No. 66 (1973), p. 1.
F. Haake, inSpringer Tracts in Modern Physics, No. 66 (1973), p. 98.
W. H. Louiseil,Quantum Statistical Properties of Radiation (Wiley, New York, 1973); M. Sargent III, M. O. Scully, and W. E. Lamb, Jr.,Laser Physics (Addison-Wesley, London, 1974).
F. Haake,Z. Physik 223:353, 364 (1969).
R. Kubo, inFluctuation, Relaxation and Resonance in Magnetic Systems, D. ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962), p. 23;Adv. Chem. Phys. 16:101 (1969), and references cited therein.
M. Tokuyama and H. Mori,Prog. Theor. Phys. 55:411 (1976).
R. J. Glauber,Phys. Rev. 131:2766 (1963); G. S. Agarwal and E. Wolf,Phys. Rev. D 2:2161, 2187, 2206 (1970).
Y. Takahashi and F. Shibata,J. Phys. Soc. Japan 38:656 (1975).
Y. Takahasi and F. Shibata,J. S tat. Phys. 14:49 (1976).
I. R. Senitzky,Phys. Rev. 115:227 (1959);119:670 (1960); R. Bonifacio and F. Haake,Z. Physik 200:526 (1967).
R. Kubo and N. Hashitsume,Prog. Theor. Phys. Suppl. 46:210 (1970); A. Kawabata,Prog. Theor. Phys. 48:2237 (1972); F. Shibata and Y. Saito,J. Phys. Soc. Japan 38:1580 (1975).
G. G. Emch and G. L. Sewell,J. Math. Phys. 9:946 (1968); G. S. Agarwal,Phys. Rev. 178:2025 (1969).
E. B. Davies,Comm. Math. Phys. 33:171 (1973);33:187 (1973); J. V. Pulé,Comm. Math. Phys. 38:241 (1974); see also L. van Hove,Physica 21:517 (1955).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hashitsumae, N., Shibata, F. & Shing¯u, M. Quantal master equation valid for any time scale. J Stat Phys 17, 155–169 (1977). https://doi.org/10.1007/BF01040099
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01040099