Invariant operators for quadratic Hamiltonians

American Journal of Physics

Volumn 67, Issue 4, 1999, Pages 336-343

  Andrews, Mark ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033421567     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19259     Document Type: Article

References (28)

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2. Eric J. Heller, "Time-dependent approach to semiclassical dynamics," J. Chem. Phys. 62, 1544-1555 (1975); V. G. Bagrov, V. V. Belov, and I. M. Ternov, "Quasiclassical trajectory-coherent states of a particle in an arbitrary electromagnetic field," J. Math. Phys. 24, 2855-2859 (1983).
3. H. R. Lewis and W. R. Riesenfeld, "An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a TimeDependent Electromagnetic Field," J. Math. Phys. 10, 1458-1473 (1969).
4. John G. Hartley and John R. Ray, "Coherent states for the time-dependent harmonic oscillator," Phys. Rev. D 25, 382-386 (1982).
5. For a recent summary, see: Michael Martin Nieto, "Displaced and squeezed number states," Phys. Lett. A 229, 135-143 (1997).
6. P. Carruthers and M. M. Nieto, "Coherent States and the Forced Quantum Oscillator," Am. J. Phys. 33, 537-544 (1965); V. V. Dodonov, I. A. Malkin, and V. I. Man'ko, "Coherent states of a charged particle in a time-dependent uniform electromagnetic field of a plane current," Physica (Amsterdam) 59, 241-256 (1972).
7. P. Carruthers and M. M. Nieto, "Coherent States and the Forced Quantum Oscillator," Am. J. Phys. 33, 537-544 (1965); V. V. Dodonov, I. A. Malkin, and V. I. Man'ko, "Coherent states of a charged particle in a time-dependent uniform electromagnetic field of a plane current," Physica (Amsterdam) 59, 241-256 (1972).
8. James Wei and Edward Norman, "Lie algebraic solution of linear differential equations," J. Math. Phys. 4, 575-577 (1963); Y.-Z. Lai, J.-Q. Liang, H. J. W. Müller-Kirsten, and Jian-Ge Zhou, "Time evolution of quantum systems with time-dependent Hamiltonian and the invariant Hermitian operator," J. Phys. A 29, 1773-1783 (1996).
9. James Wei and Edward Norman, "Lie algebraic solution of linear differential equations," J. Math. Phys. 4, 575-577 (1963); Y.-Z. Lai, J.-Q. Liang, H. J. W. Müller-Kirsten, and Jian-Ge Zhou, "Time evolution of quantum systems with time-dependent Hamiltonian and the invariant Hermitian operator," J. Phys. A 29, 1773-1783 (1996).
10. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), 2nd ed., p. 165.
11. See Ref. 7, p. 337, Eq. (15.12). Also L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, 1958), p. 9.
12. M. A. Gregório and A. S. de Castro, "A particle moving in a homogeneous time-varying force," Am. J. Phys. 52, 557-559 (1984).
13. M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, "Distribution Functions in Physics: Fundamentals," Phys. Rep. 106, 121-167 (1984), p. 140.
14. For a discussion of this for the free particle, see Hai-Woong Lee, "Spreading of a free wave packet," Am. J. Phys. 50, 438-440 (1982).
15. An alternative to this approach is to use Glauber's displacement operator. See, for example, M. Boiteux and A. Levelut, "Semicoherent states," J. Phys. A 6, 589-596 (1973).
16. See Ref. 7, p. 61.
17. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1998), 3rd ed., p. 89.
18. See Ref. 7, p. 164.
19. See, for example, B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), p. 51.
20. R. M. Cavalcanti, "The wave function of a Brownian particle," Los Alamos eprint: quant-ph/9805005, 2 May 1998. Phys. Rev. E. 58, 6807-6809 (1998).
21. E. Schrödinger, "Der stetige Übergang von der Mikro-zur Makromechanik," Naturwissenschaften 14, 664 (1926).
22. E. H. Kennard, "Zur Quantenmechanik einfacher Bewegungstypen," Z. Phys. 44, 326 (1927).
23. I. R. Senitzky, "Harmonic Oscillator Wave Functions," Phys. Rev. 95, 1115-1116 (1954).
24. A graphical representation of the behaviour of a Gaussian wave packet near a saddle point has been given by A. Askar and J. H. Weiner, "Wave Packet Dynamics on Two-Dimensional Quadratic Potential Surfaces," Am. J. Phys. 39, 1230-1234 (1970).
25. For a discussion see Max Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), p. 336.
26. For more detail on these and higher moments see: M. Andrews, "The spreading of wave packets in quantum mechanics," J. Phys. A 14, 1123-1129 (1981); and Mark Andrews and Michael Hall, "Evolution of moments over quantum wave packets or classical clusters," ibid. 18, 37-44 (1984).
27. For more detail on these and higher moments see: M. Andrews, "The spreading of wave packets in quantum mechanics," J. Phys. A 14, 1123-1129 (1981); and Mark Andrews and Michael Hall, "Evolution of moments over quantum wave packets or classical clusters," ibid. 18, 37-44 (1984).
28. For more detail on moments over sets of identical classical particles see M. Andrews, "The evolution of a cluster of particles and the invariants of a time-dependent oscillator," Phys. Lett. A 83, 95-97 (1981).