Feynman path integral, quantum

Voth G A 1993 Feynman path integral formulation of quantum mechanical transition state theory J. Phys. Chem. 97 8365... 

The path-integral quantum mechanics relies on the basic relation for the evolution operator of the particle with the time-independent Hamiltonian H x, p) = -i- V(x) [Feynman and... 

The calculation of the potential of mean force, AF(z), along the reaction coordinate z, requires statistical sampling by Monte Carlo or molecular dynamics simulations that incorporate nuclear quantum effects employing an adequate potential energy function. In our approach, we use combined QM/MM methods to describe the potential energy function and Feynman path integral approaches to model nuclear quantum effects. 

The Feynman path integral formalism " in quantum mechanics has proven to be an important vehicle for studying the quantum properties of condensed matter, both conceptually and in computational studies. Various classical-like concepts may be more easily introduced and, in the case of equilibrium properties, the formalism provides apowerful computer simulation tool. 

Our simulations are based on well-established mixed quantum-classical methods in which the electron is described by a fully quantum-statistical mechanical approach whereas the solvent degrees of freedom are treated classically. Details of the method are described elsewhere [27,28], The extent of the electron localization in different supercritical environments can be conveniently probed by analyzing the behavior of the correlation length R(fih/2) of the electron, represented as polymer of pseudoparticles in the Feynman path integral representation of quantum mechanics. Using the simulation trajectories, R is computed from the mean squared displacement along the polymer path, R2(t - t ) = ( r(f) - r(t )l2), where r(t) represents the electron position at imaginary time t and 1/(3 is Boltzmann constant times the temperature. 

G. A. Voth, Feynman Path Integral Formulation of Quantum Mechanical Transition-State Theory, J. Phys. Chem., 97 (1993) 8365. 

Many problems in D-dimensional statistical mechanics with nearest-neighbor interactions can be converted into quantum mechanics problems in (D — 1) dimensions of space and one dimension of time [84]. The quantum theory arises here in a Feynman path integral formulation [85]. 

The formulas just developed are clearly relevant to quantum dynamics, but their relevance to the Monte Carlo computation of molecular thermodynamic properties has not yet been developed. It turns out that we can develop a theory of quantum statistical mechanics [33] that is completely analogous to the Feynman path-integral version of quantum dynamics. 

In this article a perspective on quantum statistical mechanics and dynamics has been reviewed that is based on the path centroid variable in Feynman path integration [1,3-8,21-23]. Although significant progress has been achieved in this research effort to date, much remains to be done. For example, in terms of the calculation of equilibrium properties it... 

The physical adsorption isotherms on carbon materials have been studied theoretically using Grand Canonical Monte Carlo simulations and an effective classical potential [8], or using Feynmaim path formalism in conjunction with the Monte Carlo method to take into account the quantum effects [9]. To simulate hydrogen adsorption accurately at low temperature, these quantum effects have to be included. In this last case hydrogen is considered as a quantum fiuid. The basic idea of Feynman path integral formalism is to look at the possible paths that a particle can take to move from one point to another. 

The integrand represents the probability distribution of the cycle time of the thickness oscillators. In this formalism, the temporal development of the system is described in some mathematical analogy of the Feynman path integrals that also use a recursive description and probability theory [232, 233] for particle propagation in quantum electrodynamics. 

An expression for the matrix element (134.11) can be derived using the FEYNMAN path integral formulation of quantum mechanics /107/> which yields in the classical limit (h— 0)... 

From the chemically point of view, the valence states are those situated in the chemical zone -and they are the main concern forthe chemical reactivity by employing the frontier or the outer electrons consequently, the semiclas-sical approximation that models the excited states was expressly presented either as an extension of the quantum Feynman path integral or as a specialization of the Feynman-Kleinert formalism for higher temperature treatment of quantum systems (see Section 2.5). However, due to the correspondences of Table 2.1 one may systematically characterize the semiclassical (or quantum chemical) approaches as one of the limiting situations (Putz, 2009) ... 

With the quantum path decomposition (2.19) the Feynman path integral measure in Eq. (2.21) factorizes accordingly (Putz, 2009)... 

The first volume covers the concepts of nuclear, atomic, molecular and solids on the basis of quantum principles—from Planck, Bohr, Einstein, Schrodinger, Hartree-Fock, up to Feynman Path Integral approaches ... 

Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems. 

FEYNMAN S PATH INTEGRAL QUANTUM FORMALISM 4.2.1 CONSTRUCTION OF PATH INTEGRAL... 

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