Mathematical Notes

, Volume 58, Issue 6, pp 1251–1261 | Cite as

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

  • V. V. Belov
  • M. F. Kondrat'eva
Article

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 HamiltonianH(x,p), x ε ℝ N . The set of operators is chosen to be basis in the space Mat K ℂ⊗U(WN), whereU(WN) is the universal enveloping algebra of the Heisenberg-Weyl algebraWN, generated by the time-dependent operatorsÎ, x−¯x(t) · Î, andP−¯p(t) · Î, whereÎ 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 ℝ2N 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 limitħ→0 is investigated.

Keywords

Differential Equation Ordinary Differential Equation Identity Operator Poisson Bracket Momentum Operator 

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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • V. V. Belov
    • 1
  • M. F. Kondrat'eva
    • 1
  1. 1.Moscow Institute of Electronics and MathematicsUSSR

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