Evolution of States of Infinite Quantum Systems

Abstract

Consider a system of N particles contained in a finite region (vessel) ∧ with volume V - | ∧ |. A state of this system is described by a density matrix p ^∧ _N (t) that satisfies the equation

$$i\frac{{\partial \rho _N^ \wedge }}{{\partial t}} = \left[ {H_N^ \wedge ,\rho _N^ \wedge (t)} \right],\rho _N^ \wedge {(t)_{r = 0}} = \rho _N^ \wedge (0),H_N^ \wedge \equiv {H_N}( \wedge ),$$

((4.1))

with a Hamiltonian

$$H_N^ \wedge = - \sum\limits_{i = 1}^N {\left( {\frac{1}{{2m}}{\Delta _1} + {u^ \wedge }\left( {{x_i}} \right) + \mathop \sum \limits_{i < j = 1}^N \Phi \left( {{x_i} - {x_j}} \right)} \right)} ,$$

((4.2))

where u ^Λ(x) is an external field which keeps the system in the region ∧ (u ^∧(x) = 0 if x ∊ ∧ and u ^∧(x) = + ∞ if x ∉ ∧) and (φ(x) is such that the operator H ^∧ _N is selfadjoint.