Following Ehrenfest's approach, the problem of quantum-classical correspondence can be treated in the class of trajectory-coherent functions that approximate a quantum-mechanical state as → 0. This idea leads to a family of systems of ordinary differential equations, called Hamilton-Ehrenfest M-systems (M ≤ 0, 1, 2, ...). As noted in the authors' previous works, every M-system is formally equivalent to the semiclassical approximation of order M for the linear Schrödinger equation. In this paper a similar approach is undertaken for a nonlinear Hartree-type equation with a smooth integral kernel. It is demonstrated how quantum characteristics can be retrieved directly from the corresponding Hamilton-Ehrenfest systems, without solving the quantum equation: the semiclassical spectral series are obtained from the rest point solution. One of the key steps is derivation of a modified nonlinear superposition principle valid in the class of trajectory-coherent quantum states.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics