Quantum representations

Passing to the Boson operators by aid of Table II, and after neglecting the zero-point-energy of the fast mode, we obtain a quantum representation we shall name I, in which the effective Hamiltonians of the slow mode corresponding respectively to the ground and first excited states of the fast mode are... 

Although equivalent, their expressions may be more of less tractable. We shall treat them separately. In the first place, we shall consider the representation I that will have later the merit to allow us to obtain an expression for the autocorrelation function having a form similar to that of the above standard approach. Aside from this representation I, we shall also recall the other quantum representations II and III still used in the literature [8,61]. 

Note that this last expression is nothing but the closed form [90] of the autocorrelation function obtained (as an infinite sum) in quantum representation III by Boulil et al.[87] in their initial quantum approach of indirect damping. Although the small approximation involved in the quantum representation III and avoided in the quantum representation II, both autocorrelation functions are of the same form and lead to the same spectral densities (as discussed later). 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

The energy levels of a molecule placed in an off-resonant microwave field can be calculafed by diagonalizing fhe mafrix of fhe Floquef Hamiltonian in the basis of direct products y) ), where y) represents in the eigenstates of the molecule in the absence of the field and ) - fhe Fourier componenfs in Eq. (8.21). The states k) are equivalent to photon number states in the alternative formalism using the quantum representation of the field [11, 15, 26], The eigensfales of the Floquet Hamiltonian are the coherent superpositions... 

As already discussed at the end of Section 2.2.3, we derived a universal superposition principle from a complex symmetric ansatz arriving at a Klein-Gordon-like equation relevant for the theory of special relativity. This approach, which posits a secular-like operator equation in terms of energy and momenta, was adjoined with a conjugate formal operator representation in terms of time and position. As it will be seen, this provides a viable extension to the general theory [7, 82]. We will hence recover Einstein s laws of relativity as construed from the overall global superposition, demonstrating in addition the independent choice of a classical and/or a quantum representation. In this way, decoherence to classical reality seems always possible provided that appropriate operator realizations are made. 

The Adiabatic Hamiltonian and the Effective Hamiltonians Quantum representations II and III ... 

Thermal Average of the Translation Operator The Driven Damped Quantum Harmonic Oscillator Quantum Representation II ... 

Examination of Eqs. (54) shows that, within the quantum representation III, the excitation of the fast mode displaces the origin of the slow mode wave functions toward shorter lengths. That may be viewed as a translation of the slow mode potential, that is induced by the excitation of the fast mode. In order to visualize this potential displacement, it is suitable to consider the potential as Morse-like. That is depicted on the right-hand side of Fig. 2. Here, the left-hand side is devoted to the quantum representation //, where there is no potential... 

Figure 2. The equivalent quantum representations of the H-bond bridge after excitation of the fast mode.

Appendix N shows that within quantum representation /// and according to Eq. (N12), when the fast mode is in its first excited state, the ground state of the slow mode is a coherent state, that is, an eigenstate of the lowering operator ... 

Previously, we found that within the adiabatic quantum representation II, the ACF of the dipole moment operator of the fast mode may be expressed in the absence of damping according to the simple form (64), that is,... 

As a consequence of these equations, it appears that, when the fast mode is in its ground state, the slow mode does not involve any dynamics that might be the X quantum representation. 

Quantum Representation II. In quantum representation II and from Table VI, the Hamiltonian and the Boltzmann density operator of the H-bond bridge are given, respectively, by the equations ... 

In this last expression, as for the quantum representation //, the structure of a quasiclassical coherent state minimizing the uncertainty Heisenberg relations may be recognized. 

Later, it will appear that in the presence of indirect relaxation, Eqs. (91) and (94) Eqs. (92), (95), and Eqs. (93) and (96) respectively, transform into damped forms. Then, at any temperature the density operators in both quantum representations // and /// appear to be those of coherent states at any temperature (see Appendix N) ... 

Finally, the time-dependent thermal averages of the H-bond bridge coordinate, in both quantum representations II and ///, evolve according to... 

Now, let us look at the incorporation of the quantum indirect damping in the quantum representation // of the H-bond bridge. It is necessary to introduce in the model of the weak H-bond working within the strong anharmonic coupling theory, an hypothesis on the nature and on the irreversible action of... 

For this purpose, consider selective canonical transformations leading to a new quantum representation that we name /// in order to diagonalize the effective Hamiltonian corresponding to k = 1, without affecting that corresponding to k = 0. This may be performed on the different effective operators B dealing with k = 0,1 with the aid of... 

In quantum representation //, the Boltzmann density operator corresponding to the H-bond bridge viewed as a quantum harmonic oscillator may be written, neglecting the zero-point energy, according to... 

In quantum representation ///, the density operator of the slow mode corresponding to the situation where the fast mode is in its ground state 0 ) is unchanged with respect to representation // ... 

Within the adiabatic approximation, and within the quantum representation II, the Boltzmann density operator playing the physical role is... 

Now, pass to quantum representation ///. Within it, the effective Hamiltonians corresponding to the ground-state fast mode is unchanged, whereas that corresponding to the first excited state is modified. Owing to Table VI, the following equations hold ... 

The /Ith eigenstates of the harmonic Hamiltonian of the H-bond bridge in quantum representations // and ///, Franck-Condon factors. 

Figure 3 Schematic quantum representation of (a) IR absorption, (b) normal Raman scattering, and (c) RR scattering (see text). The parabolas represent harmonic potentials in the ground electronic states. The two energy levels shown in the harmonic groimd-state (GS) potentials are quantum vibrational states u = 0 is the zero-point vibrational energy, and u = 1 is the first excited vibrational state...

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