Quantum oscillator with fluctuating time-dependent frequency

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 57, Issue 4, 1998, Pages 2347-2356

  Ferrari, Loris ^a  

^a INFM   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; ELECTRODYNAMICS; ELECTROMAGNETIC FIELDS; ELECTRONS; NUMERICAL ANALYSIS; OSCILLATORS (MECHANICAL); PERTURBATION TECHNIQUES; PROBLEM SOLVING;

HAMILTONIAN; HARMONIC OSCILLATOR; NONHARMONIC INTERACTION; QUANTUM OSCILLATOR; SCHRODINGER EQUATION; VAN ALPHEN-SHUBNIKOV EFFECT;

QUANTUM THEORY;

EID: 0032046658     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.57.2347     Document Type: Article

References (22)

1. L. Ferrari, Phys. Rev. B 56, 593 (1997).PRBMDO
2. L. Landau and E. Lifshitz, Mécanique (Mir, Moscow, 1969), see footnote on pp. 214 and 215.
3. See, for example, J. M. Ziman, Principles of the Theory of Solids (Cambridge University Press, Cambridge, England, 1964), pp. 269–280;
4. C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963), pp. 217–225.
5. L. Ferrari, Phys. Rev. B (to be published).
6. See, for example, A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1970), Vol. I, pp. 210 and 211.
7. See, for example, J. Chahoud, L. Ferrari, and G. Russo, Nuovo Cimento 26, 171 (1975).
8. M. Plischke and B. Bergensen, Equilibrium Statistical Physics (World Scientific, Singapore, 1994), pp. 432–439.
9. See, for example, A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1970), Vol. II, pp. 739–755.
10. A. M. Dykhne, Zh. Eksp. Teor. Fiz. 38, 570 (1960) [Sov. Phys. JETP 11, 411 (1960)].SPHJAR
11. It should be noticed that the model fluctuation Fig. 11 is characterized by a “sudden” change, that makes the adiabatic approach somehow questionable in itself. This point, however, has nothing to do with the exponential increase of the energy, which is due to quite general reasons, independent of the specific model (for an outline, see Sec. VI). Instead, it is true that the special model Fig. 11 yields an adiabatic condition weaker than the one obtained for smoothly changing fluctuations.
12. P. W. Anderson and P. R. Weiss, Rev. Mod. Phys. 25, 269 (1954); RMPHAT
13. P. W. Anderson, J. Phys. Soc. Jpn. 9, 316 (1954); JUPSAU
14. J. Phys. Soc. Jpn.R. Kubo, 9, 935 (1954).
15. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II, Solid State Sciences Vol. 31 (Springer-Verlag, Berlin, 1978), pp. 40–48.
16. The works in Refs. 1112 are all concerned with disordered oscillations of the frequency, for which the present author predicts an exponential increase of the oscillation amplitude, as a quite universal feature, in the absence of dissipation. However, this feature is completely ignored in those works. It is reasonable that in 1954, P. W. Anderson was unaware of his own future paper [Phys. Rev. 109, 1492 (1958)] on the Anderson localization, and of its consequences in one dimension. It looks more surprising that the problem is totally ignored even in Ref. 12. We simply stress that the cornerstone of the calculations in Ref. 12 is Eq. (2.1.2) on p. 40. This motion equation, however, is not equivalent to Eq. (1), unless one discards terms proportional to (Formula presented) Discarding those terms might not be wrong, in some practical cases, such as, for example, when the dissipation effects do suppress the exponential increase (Sec. V). In short, the author’s feeling is that the works in Refs. 1112 do actually refer to such cases, though unconsciously.PHRVAO
17. P. L. Tea, Jr. and H. Falk, Am. J. Phys. 36, 1165 (1968); AJPIAS
18. Am. J. Phys.J. Burns, 38, 920 (1970);
19. Am. J. Phys.S. M. Curry, 44, 924 (1976).
20. H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).JMAPAQ
21. I. A. Malkin and V. I. Man’ko, Phys. Rev. D 2, 1371 (1970); PRVDAQ
22. I. A. MalkinV. I. Man’koPhys. Lett. 32A, 243 (1970).