The finite-time operation of a quantum-mechanical heat engine with a working fluid consisting of many noninteracting spin-l/2 systems is considered. The engine is driven by an external, time-dependent and nonrotating magnetic field. The cycle of operation consists of two adiabats and two isotherms. The analysis is based on the time derivatives of the first and second laws of thermodynamics. Explicit relations linking quantum observables to thermodynamic quantities are developed. The irreversible operation of this engine is studied in three cases: (1) The sudden limit, where the performance is found to be the same as that of the spin analog of the Otto cycle. This case provides the lower bound of efficiency. (2) The step-cycle operation scheme. Here, the optimization of power is carried out in the high-temperature limit (the "classical" limit). The results obtained are similar to those of Andresen et al. [ Phys. Rev. A 15,2086 (1977) 1. (3) The Curzon-Ahlborn operation scheme. The semigroup approach is used to model the dynamics. Then the power production is optimized. All the results obtained for Newtonian engines operating by the same scheme, such as the Curzon-Ahlborn efficiency, apply in the high-temperature limit. These results are obtained without the additional assumption of proximity to thermal equilibrium, implicitly implied by the use of Newtonian heat conduction in the original derivation. It seems that the results of the Curzon-Ahlborn analysis are always obtained in the high-temperature limit, irrespective of the details of the model. The performance beyond the classical limit is optimized numerically. The classical approximation is found to be valid for most of the spin-polarization range. The deviations from the classical limit depend heavily upon the specific nature of both the working fluid and the heat baths and exhibit great diversity and complexity.

This paper presents the first application of semiclassical methodology to the calculation of vibrational energy relaxation (VER) rate constants in condensed phase systems. The VER rate constant is treated within the framework of the Landau-Teller formula and is given in terms of the Fourier transform, at the vibrational frequency, of the force-force correlation function (FFCF). Due to the high frequency of most molecular vibrations, predictions based on the classical FFCF are often found to deviate by orders of magnitude from the experimentally observed values. In this paper, we employ a semiclassical approximation for the quantummechanical FFCF, that puts it in terms of a classical-like expression, where Wigner transforms replace the corresponding classical quantities. The multidimensional Wigner transform is performed via a novel implementation of the local harmonic approximation (LHA). The resulting expression for the FFCF is exact at t ) 0, and converges to the correct classical limit when p f 0. Quantum effects are introduced via a nonclassical initial sampling of both positions and momenta, as well as by accounting for delocalization in the calculation of the force at t ) 0. The application of the semiclassical method is reported for three model systems: (1) a vibrational mode coupled to a harmonic bath, with the coupling exponential in the bath coordinates; (2) a diatomic coupled to a short linear chain of helium atoms; (3) a "breathing sphere" diatomic in a two-dimensional monatomic Lennard-Jones liquid. Good agreement is found in all cases between the semiclassical predictions and the exact results, or their estimates. It is also found that the VER of highfrequency molecular vibrations is dominated by a purely quantum-mechanical term, which vanishes in the classical limit.

The manifestations of the three laws of thermodynamics are explored in a model of an irreversible quantum heat engine. The engine is composed of a three-level system simultaneously coupled to hot and cold heat baths, and driven by an oscillating external field. General quantum heat baths are considered, which are weakly coupled to the three-level system. The work reservoir is modeled by a classical electromagnetic field of arbitrary intensity, which is driving the three-level system. The first law of thermodynamics is related to the rate of change of energy obtained from the quantum master equation in the Heisenberg picture. The fluxes of the thermodynamic heat and work are then directly related to the expectation values of quantum observables. An analysis of the standard quantum master equation for the amplifier, first introduced by Lamb, is shown to be thermodynamically inconsistent when strong driving fields are used. A generalized master equation is rigorously derived, starting from the underlying quantum dynamics, which includes relaxation terms that explicitly depend upon the field. For weak fields the generalized master equation reduces to the standard equation. In very intense fields the amplifier splits into two heat engines. One engine accelerates as the field intensifies, while the other slows down and eventually switches direction to become a heat pump. The relative weight of the slower engine increases with the field intensity, leading to a maximum in power as a function of the field intensity. The amplifier is shown to go through four ''phases'' as the driving field is intensified, throughout all of which the second law of thermodynamics is generally satisfied. One phase corresponds to a ''refrigeration window'' which allows for the extraction of heat out of a cold bath of temperatures down to the absolute zero. This window disappears at absolute zero, which is conjectured to be a dynamical manifestation of the third law of thermodynamics.

The semiclassical theory of vibrational energy relaxation (VER) developed in the preceding paper is extended to the case of molecular liquids, and used for calculating the VER rate constant in neat liquid oxygen at 77 K. We employ a semiclassical approximation of the quantum-mechanical force-force correlation function (FFCF), which puts it in terms of the Wigner transforms of the force and the product of the Boltzmann operator and the force. The multidimensional Wigner integrals are performed via a novel implementation of the local harmonic approximation (LHA). The methodology is extended to include centrifugal forces, and effective ways are developed in order to make it applicable to molecular liquids. The fact that VER of highfrequency solutes is dominated by the solvent molecules in their close vicinity suggests that the semiclassical treatment is restricted to the small cluster of molecules around the relaxing molecule. The rest of the molecules are frozen and serve as a static cage that keeps the cluster from falling apart. The method is applied to the challenging problem of calculating the extremely slow (k 0r1 ) 395 s -1 ) and highly quantum-mechanical (pω/k B T ) 29) VER rate constant in neat liquid oxygen at 77 K. The results are found to be in very good quantitative agreement with experiment and suggest that this semiclassical approximation can capture the 4 orders of magnitude quantum enhancement of the experimentally observed VER rate constant over the corresponding classical prediction. As for the simpler models considered in the preceding paper, we find that VER in liquid oxygen is dominated by a purely quantum mechanical term, which vanishes at the classical limit. These results further establish the semiclassical method as an attractive alternative to the commonly used approach which is based on ad hoc quantum correction factors.

Quasi-classical mapping Hamiltonian methods have recently emerged as a promising approach for simulating electronically nonadiabatic molecular dynamics. The classicallike dynamics of the overall system within these methods makes them computationally feasible and they can be derived based on well-defined semiclassical approximations.However, the existence of a variety of different quasi-classical mapping Hamiltonian methods necessitates a systematic comparison of their respective advantages and limitations. Such a benchmark comparison is presented in this paper. The approaches compared include the Ehrenfest method, the symmetrical quasi-classical (SQC) method, and five variations of the linearized semiclassical (LSC) method, three of which employ a modified identity operator. The comparison is based on a number of popular nonadiabatic model systems; the spin-boson model, a Frenkel bi-exciton model and 1