Hamiltonian systems nonstationary state

Interaction with light changes the quantum state a molecule is in, and in photochemistry this is an electronic excitation. As a result, the system will no longer be in an eigenstate of the Hamiltonian and this nonstationary state evolves, governed by the time-dependent Schrodinger equation... 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

The answer is not unique, but a general statement can be made A short time external perturbation exerted on a system in a stationary state (i.e. an eigenstate of the system s Hamiltonian) will generally move the system into a nonstationary state provided that the duration of this perturbation is short relative to h/ E, where E is a typical spacing between the system s energy levels in the spectral range of interest. In what follows we describe a particular example. 

The average value of the dipole moment will be calculated by means of Dirac s perturbation theory for nonstationary. states, up to third order the zero order refers to the free molecules in the absence of the field. Let the wave function of the system of the two interacting molecules in- the external field be specified by y, an eigenfunction of the total Hamiltonian H. This wave function y> may be expanded in a complete set of the energy eigenfunctions unperturbed system the index n labels the various unperturbed eigenstates characterized by the energy En. We may then write... 

Khalfin discovered a very general result [55] that the long-time decay for Hamiltonians with spectra bounded from below is slower than exponential. The argument is this consider a system described by a time-independent Hamiltonian, H, initially in a normalized nonstationary state ho). The survival amplitude of fhat stafe is defined as the overlap of the initial state with the state at time t,... 

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