# The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians

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## Abstract

An infinite system of ordinary differential equations for*¯x, ¯p*, and for averages of a set of operators is derived for quantum-mechanical problems with a (*K×K*) matrix Hamiltonian*H(x,p), x* ε ℝ^{ N }. The set of operators is chosen to be basis in the space Mat_{ K }ℂ⊗*U(W*_{N}), where*U(W*_{N}) is the universal enveloping algebra of the Heisenberg-Weyl algebra*W*_{N}, generated by the time-dependent operators*Î, x−¯x(t) · Î*, and*P−¯p(t) · Î*, where*Î* is the identity operator and*¯x* and*¯p* are the averages of the position and momentum operators. The system in question can be written in Hamiltonian form; the corresponding Poisson bracket is degenerate and is equal to the sum of the standard bracket on ℝ^{2N} with respect to the variables (*x, p*) and the generalized Dirac bracket with respect to the other variables. The possibility of obtaining finite-dimensional approximations to the infinite-dimensional system in the semiclassical limit*ħ*→0 is investigated.

### Keywords

Differential Equation Ordinary Differential Equation Identity Operator Poisson Bracket Momentum Operator## Preview

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### References

- 1.J. von Neumann,
*Mathematische Grundlagen der Quantenmechanik*, Springer-Verlag, Berlin (1932).Google Scholar - 2.F. A. Berezin and M. A. Shubin,
*The Schrödinger Equation*[in Russian], Izd. Moskov. Univ., Moscow (1983).Google Scholar - 3.V. V. Belov and M. F. Kondrat'eva “Hamiltonian systems of equations for quantum averages,”
*Mat. Zametki*[*Math. Notes*],**56**, No. 6, 27–39 (1994).MathSciNetGoogle Scholar - 4.M. V. Karasev and V. P. Maslov,
*Nonlinear Poisson Brackets*[in Russian], Nauka, Moscow (1991).Google Scholar - 5.M. F. Kondrat'eva,
*Semiclassical Trajectory-Coherent States and the Evolution of Quantum Averages*[in Russian], Candidate of Sciences Thesis, Tomsk University (1993).Google Scholar - 6.V. G. Bagrov, V. V. Belov and M. F. Kondrat'eva, “Semiclassical approximation in quantum mechanics,”
*Teor. Mat. Fiz.*[*Theoret. and Math. Phys.*],**98**, No. 1, 48–55 (1994).MathSciNetGoogle Scholar - 7.V. G. Bagrov, V. V. Belov, M. F. Kondrat'eva, A. M. Rogova, and A. Yu. Trifonov, “The quasiclassical localization of the states and new approach to quasiclassical approximation in quantum mechanics,” in:
*Proc. of the 5th and 6th Lomonosov Conf. on Elementary Particles Physics “Particle Physics, Gauge Fields and Astrophysics”*, Accademia Nazionale dei Lincei, Rome (1994).Google Scholar - 8.V. G. Bagrov, V. V. Belov, M. F. Kondrat'eva, A. M. Rogova, and A. Yu. Trifonov, “A new formulation of quasi-classical approximation in quantum mechanics,”
*J. Moscow Phys. Soc.*,**3**, 1–12 (1993).MathSciNetGoogle Scholar - 9.V. G. Bagrov, V. V. Belov, and A. Yu. Trifonov, “A new method for semiclassical approximation in quantum mechanics,” in:
*Proceedings of the International Conference “Geometrization of Physics”*[in Russian], Remark, Kazan (1994), pp. 66–77.Google Scholar - 10.V. G. Bagrov, V. V. Belov, and A. M. Rogova, “Semiclassically concentrated states in quantum mechanics,”
*Teor. Mat. Fiz. [Theoret. and Math. Phys.]*,**90**, No. 1 (1992).Google Scholar - 11.V. G. Bagrov, V. V. Belov, A. M. Rogova, and A. Yu. Trifonov, “The quasiclassical localization of the states and obtaining of classical equations of motion from quantum theory”
*Modern Phys. Lett.*,**7**, No. 26, 1667–1675 (1993).Google Scholar - 12.V. G. Bagrov, V. V. Belov, and A. Yu. Trifonov, “Higher approximations for semiclassical trajectory-coherent states of the Schrödinger and Dirac operators in an arbitrary electromagnetic field,”
*Preprint No.242 5*[in Russian], Tomsk Scientific Center of the Siberian Division of the Russian Academy of Sciences, Tomsk (1993).Google Scholar - 13.V. V. Kucherenko, “Asymptotic solutions to the system
*A(x, -iħ∂*_{x})u=0 for*ħ*→0 in the case of characteristics of variable multiplicity,”*Izv. Akad. Nauk SSSR, Ser. Mat.*[*Math. USSR-Izv.*],**38**, No. 3, 625–662 (1974).MATHMathSciNetGoogle Scholar - 14.V. V. Kucherenko and Yu. V. Osipov, “Asymptotics of the Cauchy problem for nonstrictly hyperbolic equations,”
*Mat. Sb.*[*Math. USSR-Sb.*],**120**, No. 1, 84–111 (1983).MathSciNetGoogle Scholar - 15.A. M. Perelomov,
*Integrable Systems of Classical Mechanics and Lie Algebras*[in Russian], Nauka, Moscow (1990).Google Scholar - 16.V. V. Belov, “Semiclassical limit for equations of motion for quantum averages for nonrelativistic systems with gauge fields,”
*Preprint No.242 58*[in Russian], Siberian Division of the Russian Academy of Sciences, Tomsk (1989), p. 30.Google Scholar - 17.V. V. Belov,
*Semiclassical Trajectory-Coherent Approximation in Quantum Theory*[in Russian], Doctor of Sciences Thesis, Moscow State University, Moscow (1991).Google Scholar - 18.V. V. Belov and V. P. Maslov, “Semiclassical trajectory-coherent states in quantum mechanics with gauge fields,”
*Dokl. Akad. Nauk SSSR*[*Soviet Math. Dokl.*],**311**, No. 4, 849–854 (1990).MathSciNetGoogle Scholar - 19.V. V. Belov and M. F. Kondrat'eva, ““Classical” equations of motion in quantum mechanics with gauge fields,”
*Teor. Mat. Fiz.*[*Theoret. and Math. Phys.*],**92**, No. 1, 41–64 (1992).MathSciNetGoogle Scholar - 20.L. I. Schiff,
*Quantum Mechanics*, 2nd ed., McGraw-Hill, New York (1955).Google Scholar - 21.K. Huang,
*Quarks, Leptons, and Gauge Fields*, World Scientific, Singapore (1982).Google Scholar - 22.Ya. I. Frenkel,
*Collected Papers*[in Russian], Vol. 2, Izd. AN SSSR, Moscow-Leningrad (1958).Google Scholar - 23.S. K. Wong, “Fields and particle equation for the classical Yang-Mills field and particles with isotopic spin,”
*Nuovo Cim.*,**A65**, No.4, 689–694 (1970).Google Scholar