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

An infinite system of ordinary differential equations for $\bar x$, $\bar p$, and for averages of a set of operators is derived for quantum-mechanical problems with a $(K\times K)$ matrix Hamiltonian $\mathscr H(\hat x,\hat p)$, $x\in\mathbb R^N$. The set of operators is chosen to be basis in the space $\mathrm{Mat}_K\mathbb C\otimes U(\mathscr W_N)$, where $U(\mathscr W_N)$ is the universal enveloping algebra of the Heisenberg–Weyl algebra $\mathscr W_N$, generated by the time-dependent operators $\hat I$, $\hat x-\bar x(t)\cdot\hat I$, $\hat p-\bar p(t)\cdot\hat I$, where $\hat I$ is the identity operator and $\bar x$, $\bar p$ 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 $\mathbb R^{2N}$ with respect to the variables $(\bar x,\bar p)$ 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 $\hbar\to0$ is investigated.