Prog. Theor. Phys. Vol. 37 No. 6 (1967) pp. 1080-1099
On the Propagation of Quantum Wave
— Generalized Random Phase Approximation
Department of Physics, Tokyo University of Education, Tokyo
(Received February 10, 1967)
The time development of the Heisenberg operator is analyzed by separating the total system into two parts in such a way that one contains the operator and the other does not. The part which does not contain the operator is assumed to be known completely. An approximation is introduced which yields exact excitation spectrum except for the O(1/Ω) (Ω being volume) correction of elementary excitations in lowest order. The formulation of growing wave of light is taken as an example, and it is reasonably argued that in the case where a single quantum state is occupied by finite density of quanta the approximation we have introduced is the expansion in terms of density of quanta.
DOI : 10.1143/PTP.37.1080
See for example, Lamb, Phys. Rev. 134 (1964), A1429[APS].
Shen and Bloembergen, Phys. Rev. 137 (1965), A1787[APS].
Bloembergen, Non-linear Optics (W.A. Benjamin, Inc., 1964).
- For the random phase approximation, see for example; Pines, The Many Body Problem; Pines and Nojieres, Quantum Liquid, I (W.A. Benjamin, Inc., 1961 and 1966).
- A similar situation holds in plasma instability;
Pines and Schrieffer, Phys. Rev. 124 (1961), 1387[APS].
Ambegaokar and Kohn, Phys. Rev. 117 (1960), 423[APS].
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ibid. 75 (1949), 486[APS];
ibid. 75 (1949), 1736[APS].
Bloembergen and Shen, Phys. Rev. 141 (1966), 298[APS].
Shen and Bloembergen, Phys. Rev. 137 (1965), 1787[APS].
Bloembergen, Non-Linear Optics (W.A. Benjamin, Inc., 1964).
Nishikawa and Takano, J. Phys. Soc. Jpn. 22 (1967), 1446[JPSJ].