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

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 ^N. The set of operators is chosen to be basis in the space Mat_KU(W_N), whereU(W_N) is the universal enveloping algebra of the Heisenberg-Weyl algebraW_N, 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 ^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 limit0 is investigated.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 803–817, December, 1995.This research was supported by the International Science Foundation under grant No. 78000.