Quantum bath

For a system of harmonic oscillators it is easy to derive the quantum equivalents of the results obtained above, as was already exemplified by Problem 6.6(3). By way of demonstration we focus on the time correlation function, Caa I) = 

Cy (wy(/)My(0)). The normal mode position operator can be written in terms of the raising and lowering operators (cf. Eq. (2.153) note that the mass factor does 

Cab(S ) = ih/7t) coJ co), demonstrating that the identity (6.35) does not hold for quantum correlation functions. 

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The difficulty in simulating the full quantum dynamics of large many-body systems has stimulated the development of mixed quantum-classical dynamical schemes. In such approaches, the quantum system of interest is partitioned into two subsystems, which we term the quantum subsystem, and quantum bath. Approximations to the full quantum dynamics are then made such that... 

NON-MARKOVIAN QUANTUM BATH AND FLUCTUATION-DISSIPATION THEOREM... 

This article will focus on the reduced dynamics formulations for a general quantum system (ifg) embedded in a dissipative quantum bath (/ib) in the presence of time-dependent external classical field e t). Denote the system-... 

The canonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same temperature T. This corresponds to putting the systems in a thennostatic bath or, since the number of systems is essentially infinite, simply separating them by diathennic walls and letting them equilibrate. In such an ensemble, the probability of finding the system in a particular quantum state / is proportional to where UfN, V) is tire energy of the /th quantum state and /c, as before, is the Boltzmaim... 

VER occurs as a result of fluctuating forces exerted by the bath on the system at the system s oscillation frequency O [5]. Fluctuating dynamical forces are characterized by a force-force correlation function. The Fourier transfonn of this force correlation function at Q, denoted n(n), characterizes the quantum mechanical frequency-dependent friction exerted on the system by the bath [5, 8]. 

Mavri, J., Berendsen, H.J.C. Dynamical simulation of a quantum harmonic oscillator in a noble-gas bath by density matrix evolution. Phys. Rev. E 50 (1994) 198-204. 

In 1979, a viable theory to explain the mechanism of chromium electroplating from chromic acid baths was developed (176). An initial layer of polychromates, mainly HCr3 0 Q, is formed contiguous to the outer boundary of the cathode s Helmholtz double layer. Electrons move across the Helmholtz layer by quantum mechanical tunneling to the end groups of the polychromate oriented in the direction of the double layer. Cr(VI) is reduced to Cr(III) in one-electron steps and a colloidal film of chromic dichromate is produced. Chromous dichromate is formed in the film by the same tunneling mechanism, and the Cr(II) forms a complex with sulfate. Bright chromium deposits are obtained from this complex. 

The most general problem should be that of a particle in a nonseparable potential, linearly coupled to an oscillator heat bath, when the dynamics of the particle in the classically accessible region is subject to friction forces due to the bath. However, this multidimensional quantum Kramers problem has not been explored as yet. 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

The classical bath sees the quantum particle potential as averaged over the characteristic time, which - if we recall that in conventional units it equals hjk T- vanishes in the classical limit h- Q. The quasienergy partition function for the classical bath now simply turns into an ordinary integral in configuration space. 

Except for the nonlocal last term in the exponent, this expression is recognized as the average of the one-dimensional quantum partition function over the static configurations of the bath. This formula without the last term has been used by Dakhnovskii and Nefedova [1991] to handle a bath of classical anharmonic oscillators. The integral over q was evaluated with the method of steepest descents leading to the most favorable bath configuration. 

The analytic results for the spin-boson Hamiltonian with fluctuating tunneling matrix element (5.67) are investigated in detail by Suarez and Silbey [1991a]. Here we discuss only the situation when the qi vibration is quantum, i.e., (o P P 1. When the bath is classical, cojP, j 1, the rate... 

At low temperatures, when the bath is quantum (icoj P 1), the rate expression, expanded in series over the coupling strength, breaks up into the contributions from the various processes involving the bath phonons... 

As discussed above, a cmcial aspect is the interaction of the reactant with the solvent. In a quantum theory, the solvent can be represented as a bath of harmonic... 

Although Eqs. (4-1) and (4-2) have identical expressions as that of the classical rate constant, there is no variational upper bound in the QTST rate constant because the quantum transmission coefficient Yq may be either greater than or less than one. There is no practical procedure to compute the quantum transmission coefficient Yq- For a model reaction with a parabolic barrier along the reaction coordinate coupled to a bath of harmonic oscillators, the quantum transmission... 

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