Abstract
We give a brief review of the theory of quantum universal invariants and their counterparts in the physics of light and particle beams. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, or the transverse phase-space coordinates of the paraxial beams of light or particles. They are conserved in time (or along the beam axis) independently of the concrete form of the coefficients of the Schrödinger-like equations governing the evolution of the systems, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation of the quantum (optical) invariants to the classical universal integral invariants of Poincaré and Cartan. The specific features of Gaussian beams are discussed as examples. The concept of the universal quantum integrals of motion is introduced, and examples of the “universal invariant solutions” to the Schrödinger equation, i.e., self-consistent eigenstates of the universal integrals of motion, are given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. R. Lewis, Jr. and W. B. Riesenfeld,J. Math. Phys.,10, 1458 (1969).
I. A. Malkin, V. I. Man'ko, and D. A. Trifonov,Phys. Lett. A,30, 414 (1969).
I. A. Malkin and V. I. Man'ko,Phys. Lett. A,32, 243 (1970).
I. A. Malkin, V. I. Man'ko, and D. A. Trifonov,Phys. Rev. D,2, 1371 (1970).
V. V. Dodonov, I. A. Malkin, and V. I. Man'ko,Physica,59, 241 (1972).
I. A. Malkin, V. I. Man'ko, and D. A. Trifonov,J. Math. Phys.,14, 576 (1973).
V. V. Dodonov, I. A. Malkin, and V. I. Man'ko,Int. J. Theor. Phys.,14, 37 (1975).
V. V. Dodonov, I. A. Malkin, and V. I. Man'ko,Physica A,82, 113 (1976).
V. V. Dodonov and V. I. Man'ko,Phys. Rev. A,20, 550 (1979).
V. V. Dodonov and V. I. Man'ko,Invariants and Evolution of Nonstationary Quantum Systems, Proceedings of the P. N. Lebedev Physical Institute, Nauka, Moscow (1987), Vol. 183 [translated by Nova Science, Commack, New York, (1989)].
F. Gantmacher,Lectures on Analytical Mechanics [Russian translation], Mir, Moscow (1970).
V. I. Arnold,Mathematical Methods of Classical Mechanics, Springer, New York (1978).
V. V. Dodonov and V. I. Man'ko, “Universal invariants of quantum systems and generalized uncertainty relations” in: M. A. Markov, V. I. Man'ko, and A. E. Shabad (eds.),Group Theoretical Methods in Physics, Proceedings of the Second International Seminar (Zvenigorod Moscow Region, November 1982), Nauka, Moscow (1983), Vol. 2, p. 11 [translated by Harwood Academic Publishers, Chur-London-Paris-New York (1985), Vol. 1, p. 591].
B. Remaud and E. S. Hernández,Phys. Lett. A,75, 269 (1980).
R. E. Turner and R. F. Snider,Can. J. Phys.,59, 457 (1981).
V. V. Dodonov and O. V. Man'ko, “Universal invariants of paraxial optical beams,” in: V. V. Dodonov, M. A. Markov, and V. I. Man'ko (eds.),Group Theoretical Methods in Physics, Proceedings of the Third International Seminar (Yurmala USSR, May 1985), VNU Science Press, Utrecht (1986), Vol. 2, p. 523.
V. V. Dodonov and O. V. Man'ko, “Universal invariants of the paraxial optical beams,” in: E. P. Velikhov and A. M. Prokhorov (eds.),Computing Optics. Fundamentals [in Russian], International Center of Scientifical and Technical Information, Institute of General Physics, and Institute of the Problems of Information Transfer, Moscow (1987), p. 84.
V. V. Dodonov and V. I. Man'ko, “Density matrices and Wigner functions of quasiclassical quantum systems,” in:Group Theory, Gravitation and Elementary Particle Physics, Proceedings of the P. N. Lebedev Physical Institute, Nauka, Moscow (1986), Vol. 167, p. 7 [translated by Nova Science, Commack, New York (1987), p. 7].
N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, and K.-B. Wolf,Phys. Lett. A,215, 128 (1996).
A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K.-B. Wolf,Phys. Rev. A,55, 876 (1997).
A. J. Dragt, R. L. Gluckstern, F. Neri, and G. Rangarajan, “Theory of emittance invariance,” in: M. Month and S. Turner (eds.),Frontiers of Particle Beams; Observation, Diagnosis, and Correction, Proceedings of the Joint USA-CERN School on Particle Accelerators (Anacapri, Italy, October 1988), Springer, Berlin (1989), p. 94.
D. D. Holm, W. P. Lysenko, and J. C. Scovel,J. Math. Phys.,31, 1610 (1990).
F. Neri and G. Rangarajan,Phys. Rev. Lett.,64, 1073 (1990).
A. J. Dragt, F. Neri, and G. Rangarajan,Phys. Rev. A,45, 2572 (1992).
M. J. Bastiaans,Optik,88, 163 (1991).
M. Kauderer,Appl. Opt.,30, 1025 (1991).
J. Serna, R. Martínez-Herrero, and P. M. Mejías,J. Opt. Soc. Am. A,8, 1094 (1991).
P. A. Belanger,Opt. Lett.,16, 196 (1991).
R. Martínez-Herrero, P. M. Mejías, M. Sanchez, and J. L. H. Neira,Opt. Quantum Electron.,24, S1021 (1992).
D. Onciul,J. Opt. Soc. Am. A,10, 295 (1993).
Q. Lin, S. Wang, J. Alda, and E. Bernabeu,Opt. Lett.,18, 669 (1993).
G. Nemes and A. E. Siegman,J. Opt. Soc. Am. A,11, 2257 (1994).
D. Dragoman,J. Opt. Soc. Am. A,11, 2643 (1994).
D. Dragoman,Appl. Opt.,34, 3352 (1995).
R. Martínez-Herrero and P. M. Mejías,Opt. Commun.,140, 57 (1997).
J. Yang and D. Fan,J. Opt. Soc. Am. A,16, 2488 (1999).
F. R. Gantmacher,The Theory of Matrices [in Russian], Nauka, Moscow (1966).
S. Chountasis and A. Vourdas,Phys. Rev. A,58, 1794 (1998).
V. V. Dodonov, V. I. Man'ko, and V. N. Rudenko,Kvantovaya Élektron.,7, 2124 (1980). [Sov. J. Quant. Electron.,10, 1232 (1980)].
V. I. Tatarskii,Usp. Fiz. Nauk,139, 587 (1983) [Sov. Phys. Usp.,26, 311 (1983)].
M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner,Phys. Rep.,106, 121 (1984).
M. J. Bastiaans,J. Opt. Soc. Am.,69, 1710 (1979).
R. G. Littlejohn,Phys. Rep.,138, 193 (1986).
R. Simon, E. C. G. Sudarshan, and N. Mukunda,Phys. Rev. A,36, 3868 (1987).
V. V. Dodonov, O. V. Man'ko, and V. I. Man'ko,Phys. Rev. A,50, 813 (1994).
V. V. Dodonov, E. V. Kurmyshev, and V. I. Man'ko,Phys. Lett. A,79, 150 (1980).
V. V. Dodonov, O. V. Man'ko, and V. I. Man'ko,Phys. Rev. A,49, 2993 (1994).
J. Snygg,Am. J. Phys.,48, 964 (1980).
T. Takabayasi,Progr. Theor. Phys.,11, 341 (1954).
H. P. Robertson,Phys. Rev.,46, 794 (1934).
V. V. Dodonov, O. V. Man'ko, and V. I. Man'ko,J. Russ. Laser Res.,16, 1 (1995).
G. Dattoli, C. Mari, M. Richetta, and A. Torre,Nuovo Cim. B,107, 269 (1992).
E. Schrödinger,Ber. Kgl. Akad. Wiss. Berlin,24, 296 (1930).
H. P. Robertson,Phys. Rev.,35, 667 (1930).
B. Remaud and S. Hernandez,Physica A,103, 35 (1980).
E. A. Akhundova, V. V. Dodonov, and V. I. Man'ko,Teor. Mat. Fiz.,60, 413 (1984) [Theor. Math. Phys.,60, 907 (1985)].
V. V. Dodonov and V. I. Man'ko,Physica A,137, 306 (1986).
V. V. Dodonov and V. I. Man'ko,Physica A,94, 403 (1978).
A. O. Barut and R. Rączka,Theory of Group Representations and Applications, World Scientific, Singapore (1986).
V. V. Dodonov, V. I. Man'ko, and D. V. Zhivotchenko,Nuovo Cim. B,108, 1349 (1993).
M. A. Leontovich,Izv. Akad. Nauk SSSR,8, 16 (1944).
M. A. Leontovich and V. A. Fock,Zh. Éksp. Teor. Fiz.,16, 557 (1946).
J. A. Arnaud,Beam and Fiber Optics, Academic, New York (1976).
R. Simon, N. Mukunda, and E. C. G. Sudarshan,Opt. Commun.,65, 322 (1988).
E. Collett and E. Wolf,Opt. Commun.,32, 27 (1980).
R. Simon, E. C. G. Sudarshan, and N. Mukunda,Phys. Rev. A,29, 3273 (1984).
R. Simon, E. C. G. Sudarshan, and N. Mukunda,Phys. Rev. A,31, 2419 (1985).
M. W. Sasnett, “Propagation of multimode laser beams—theM 2 factor,” in: D. R. Hall and P. E. Jackson (eds.),Physics and Technology of Laser Resonators, Hilger, Bristol (1989), p. 132.
A. E. Siegman, “New developments in laser resonators,” in: D. A. Holmes (ed.),Optical Resonators, Proc. SPIE,1224, 2 (1990).
M. W. Sasnett and T. F. Johnston, Jr., “Beam characterization and measurement of propagation attributes,” in:Laser Beam Diagnostics, Proc. SPIE,1414, 21 (1991).
A. E. Siegman,IEEE J. Quantum Electron.,QE-27, 1146 (1991).
T. F. Jonston, M. W. Sasnett, J. L. Doumont, and A. E. Siegman,Opt. Lett.,17, 198 (1992).
M. A. Porras,Opt. Commun.,109, 5 (1994).
H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden,IEEE J. Quantum Electron.,QE-32, 400 (1996).
C. Martínez, F. Encinas-Sanz, J. Serna, P. M. Mejías, and R. Martínez-Herrero,Opt. Commun.,139, 299 (1997).
F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero, and P. M. Mejías,IEEE J. Quantum Electron.,QE-34, 1835 (1998).
K.-B. Wolf,Opt. Commun.,132, 343 (1996).
J. Paye and A. Migus,J. Opt. Soc. Am. B,12, 1480 (1995).
Q. Cao and X. M. Deng,Opt. Commun.,142, 135 (1997).
A. E. Siegman,Appl. Opt.,32, 5893 (1993).
R. Fedele and G. Miele,Nuovo Cim. D,13, 1527 (1991).
G. Dattoli, L. Giannessi, C. Mari, M. Richetta, and A. Torre,Opt. Commun.,87, 175 (1992).
R. Fedele, F. Galluccio, V. I. Man'ko, and G. Miele,Phys. Lett. A,209, 263 (1995).
R. Fedele and V. I. Man'ko,Phys. Rev. E,58, 992 (1998).
R. Fedele, M. A. Man'ko, and V. I. Man'ko,J. Russ. Laser Res.,21, 1 (2000).
V. V. Dodonov and A. B. Klimov,Phys. Rev. A,53, 2664 (1996).
K. E. Cahill and R. J. Glauber,Phys. Rev.,177, 1882 (1996).
A. Papoulis,J. Opt. Soc. Am.,64, 779 (1974).
S. Chountasis and A. Vourdas,J. Phys. A,32, 6949 (1999).
V. V. Dodonov and O. V. Man'ko,Physica A,130, 353 (1985).
S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii,Introduction to Statistical Radiophysics [in Russian], Nauka, Moscow (1978).
Author information
Authors and Affiliations
Additional information
Manuscript submitted by the authors in English on August 2, 2000.
Rights and permissions
About this article
Cite this article
Dodonov, V.V., Man'ko, O.V. Universal invariants in quantum mechanics and physics of optical and particle beams. J Russ Laser Res 21, 438–464 (2000). https://doi.org/10.1007/BF02508736
Issue Date:
DOI: https://doi.org/10.1007/BF02508736