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Evolution of States of Infinite Quantum Systems

  • D. Ya. Petrina
Chapter
  • 334 Downloads
Part of the Mathematical Physics Studies book series (MPST, volume 17)

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.

Keywords

Canonical Ensemble Selfadjoint Operator Infinitesimal Generator Grand Canonical Ensemble Gibbs Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • D. Ya. Petrina
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKievUkraine

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