A simple quantum-mechanical model for relaxation

In what follows we consider a simple quantum-mechanical model forirreversibihty. In addition to providing a simple demonstration of how irreversibility arises in quantum mechanics, we will see that this model can be used as a prototype of many physical situations, showing not only the property of irreversible relaxation but also many of its observable consequences. 

Another way to express this information is by writing the matrix representation of H in the given basis. 

We want to solve the time-dependent Schrodinger equation 

Before setting to solve this mathematical problem, we should note that while the model is mathematically sound and the question asked is meaningfiil, it cannot represent a complete physical system. If the Hamiltonian was a real representation of a physical system we could never prepare the system in state 11). Still, we shall see that this model represents a situation which is ubiquitous in molecular systems, not necessarily in condensed phase. Below we outline a few physical problems in which our model constitutes a key element  

Consider the generic two-level model, Eq. (2.13), with the levels now denoted g and 5 with energies Eg Eg. An extended system that includes also the environment may be represented by states that will be denoted s, e ), g, e ) where e defines states of the environment. A common phrase is to say that these molecular states are dressed by the environment. Now consider this generic molecule in state s and assume that the environment is at zero temperature. In this case e = e g is the ground state of the environment. Obviously the initial state , e g) is energetically embedded in a continuum of states g, e x) where e x are excited states of the environment. This is exactly the situation represented in F ig. 9.1, where level 11) represents the state 5, e g) while levels /) are the states g, e x) with different excited state of the environment. An important aspect common to all models of this type is that the continuous manifold of states /) is bound from below State Ig, e g) is obviously its lowest energy state. 

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I shall first discuss briefly the experimental techniques Involved. I shall then review the effects of bond dissociation anharmonlclty in a diatomic molecule. Next I shall introduce the idea of local modes with a simple classical model, and then extend this to a mathematically defined quantum mechanical model which I shall discuss in detail for the case of two symmetry related stretching vibrations, as in the water molecule. I shall then introduce the effects of Fermi resonance, and describe some of our recent work on the dlchloromethane molecule. I shall also describe similar fits to the overtones of carbonyl stretching vibrations in metal carbonyls. Finally I shall comment briefly on the implications of this work for intramolecular vibrational relaxation (IVR) and chemical dynamics. 

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