Quantum degrees

Blake N P and Metiu H 1995 Efficient adsorption line shape calculations for an electron coupled to many quantum degrees of freedom, applications to an electron solvated in dry sodalites and halo-sodalites J. Chem. Phys. 103 4455... 

The reason that non-adiabatic transitions must be included for protons is that fluctuations in the potential for the quantum degrees of freedom due to the environment (e.g. solvent) contain frequencies comparable to the transition frequencies between protonic quantum states. In such cases pure quantum states do not persist. 

There are various approaches to the problem of coupling quantum degrees of freedom to classical degrees of freedom. The QCMD model is given by the following equations of motion ... 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

For ease of presentation, we consider the case of just one quantum degree of freedom with spatial coordinate x and mass m and N classical particles with coordinates q e and diagonal mass matrix M e tj Wxsjv Upon... 

The computer simulation of models for condensed matter systems has become an important investigative tool in both fundamental and engineering research [149-153] for reviews on MC studies of surface phenomena see Refs. 154, 155. For the reahstic modeling of real materials at low temperatures it is essential to take quantum degrees of freedom into account. Although much progress has been achieved on this topic [156-166], computer simulation of quantum systems still lags behind the development in the field of classical systems. This holds particularly for the determination of dynamical information, which was not possible until recently [167-176]. 

The quantity Ef is the energy of the reorganization of all the classical degrees of freedom of the local vibrations and of the classical part of the medium polarization, and crc is the tunneling factor for quantum degrees of freedom 1) which do not... 

We now consider a simple model of position measurement to provide a measure of concreteness. In this model, we will assume that there are no environmental channels aside from those associated with the measurement. Suppose we have a single quantum degree of freedom, position in this case, under a weak, ideal continuous measurement (C.M. Caves et.al., 1987). Here ideal refers to no loss of information during the measurement, i.e., a fine-grained evolution with no loss of unitarity. Then, we have two coupled equations, one for the measurement record y(t),... 

In a mixed quantum-classical calculation the trace operation in the Heisenberg representation is replaced by a quantum-mechanical trace (tTq) over the quantum degrees of freedom and a classical trace (i.e., a phase-space integral over the initial positions xq and momenta Po) over the classical degrees of freedom. This yields... 

As explained in the Introduction, most mixed quantum-classical (MQC) methods are based on the classical-path approximation, which describes the reaction of the quantum degrees of freedom (DoF) to the dynamics of the classical DoF [9-22]. To discuss the classical-path approximation, let us first consider a diabatic... 

Let us now consider the system with an arbitrary spectrum of normal vibrations. In this case normal vibrations should be first divided into classical < T) and quantum (cok > T) vibrations. If the reorganization energy of the classical vibrations exceeds the reaction exothermicity then, neglecting the excitation and absorption of phonons with the frequencies k > T, in the same way as when deriving eqn. (41), i.e. taking into account, for the quantum degrees of freedom, only the transitions (0 - 0), we obtain for the probability of tunneling the expression... 

The system is defined as the set of those quantum degrees of freedom that one is interested to control and measure the environment consists of all the rest, namely those degrees-of-freedom we can neither control nor measure. The coupling between the system and environment is VCOUpiing- The properties of the environment are controlled by macroscopic parameters, such as temperature. Our treatment below applies to a reservoir at either zero or a finite temperature. 

In many cases, in order to compute the dynamics of condensed phase systems, one invokes a basis representation for the quantum degrees of freedom in the system. Typically, one computes the dynamics of these systems in order to obtain quantities of interest, such as an average value, A(t) = Tr [Ap(t)], or a correlation function, as will be discussed below. Since such averages are basis independent one may project Eq. (8) onto any convenient basis. This is in principle a nice feature, and one that is often exploited to aid in calculations. However, it is important to note that the basis onto which one chooses to project the QCLE has important implications on how one goes about solving the resulting equations of motion. Ultimately the time-dependent average value of an observable is expressed as a trace over quantum subsystem... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

In summary, we have shown in this section that even in one of the simplest conceivable models for a diatomic molecule, the coupling of classical and quantum degrees of freedom can lead to genuine chaos in both the quantum and the classical subsystems. This proves the existence of type II quantum chaos. This result is of general importance since type II quantum chaos may occur in any system that divides in a natural way into a... 

Classical molecular dynamics simulations with quantum degrees of freedom... 

Although the wave packet description of the classical modes is definitely much better than any classical trajectory description of the non-quantum degrees of freedom, it is only applicable (without further restrictions) to systems with a very limited number of modes. This is due to the nonlinearity in the equations of motion (11-15). One possibility to overcome these difficulties is the classically based separable potential (CSP) approach which has recently suggested by Gerber and co-workers [49, 50, 51]. The method will be outlined in subsection 3.3. Before doing this, a even more simplifying approximation is described in which the non-quantum part of the model is represented by a single classical trajectory. 

In order to obtain a more compact formulation of the mixed quantum-classical equations we use a Hamilton-Jacobi-like formalism for the propagation of the quantum degree of freedom as in earlier studies [23], A similar approach has been introduced by Nettesheim, Schiitte and coworkers [54, 55, 56], TTie formalism presented here is based on recent investigations of the present authors [23], This formalism can be summarized as follows. Starting from the Hamiltonian Eqn. (2.2) and averaging over the x- and y-mode, respectively, gives... 

Another reduction is possible when one wishes to treat only one or a few degrees of freedom quantum-mechanically while the rest of the system can be treated still in a classical way. First pioneering studies along such lines treated the problem of electron solvation in molten salts and liquid ammonia. But, it must be noted that, when one studies the dynamics of quantum degrees of freedom coupled to a classical environment, particular care is required This mixed quantum-classical dynamics has subtle features, and is still an active area of research. 

Although the monochromatic plane waves of photons are described by only two quantum numbers, specifying the polarization, the monochromatic multipole waves of photons have much more quantum degrees of freedom the type of radiation (parity) X = E,M and the angular momentum j > 1 and its projection m = j,..., /. 

If only the perpendicular vibrational motion is quantized, then the operator-algebraic treatment of the quantum mechanical part of the system can be introduced and, hence, the effective Hamiltonian, which couples the reaction path motion and the quantum degrees of freedom, can be given a very compact form. Thus, the effective Hamil-... 

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