Time evolution of a two state quantum system by a norm preserving, non-Hermitian Hamiltonian

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1974

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Abstract

The Schroedinger equation is used to describe a quantum system evolving in time by a non-Hermitian Hamiltonian. It is shown that norm can be preserved if the non-Hermitian part of the Hamiltonian, H^, evolves in the interaction picture by ([partial derivative]/[parital derivative][raised T])H[lowered b]=-2H[lowered b]H[lowered b]-BH[lowered b] where B is a real scalar. It is demonstrated that this description can model the spontaneous approach to equilibrium of a thermodynamic system with B and H[lowered b] describing the coupling of the system to the environment. An explicit relation of B and H[lowered b] to the temperature and energy levels of the system is obtained for the case when the Hermitian and non-Hermitian parts of the Hamiltonian commute. A general two state system is examined by numerical integration and found to be so restrictive because of the two states that it would not be useful for describing spontaneous systems. Because of the success of the many state commuting case, a general solution of a many state system would probably be more useful than the restrictive two state system for describing spontaneous thermodynamic behavior.

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