Feynman path integral, quantum theory

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

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. 

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. 

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. 

In di.scussing these problems, it is, of course, important and natural to indicate the parallels in fields other than polymer chemistry and physics. Thus it is natural to discuss briefly any connections or similarities with Feynman path integrals, with descriptions of Brownian motion and other random processes, " and with the functional integral formulation of many body or quantum field theories. Only the most rudimentary... 

Following Feynman s original work, several authors pursued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g.. Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed," which gave rise to path integral quantum transition state theory" and even more general approaches." The Centroid Molecular Dynamics (CMD) method" for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the effective centroid potential. Various analysis" " and numerical tests for realistic systems" have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

The semiclassical theories of Gutzwiller (1967, 1971, 1980), Balian and Bloch (1972), and Berry and Tabor (1976, 1977), which are based on Feynman s path-integral formulation of quantum mechanics (Feynman and Hibbs 1965), provide a solid theoretical justification. 

I believe [2] that the fundamental original description is the Feynman s continual integral in quantum field theory [3] (which is similar to the path integral in quantum mechanics). 

In the preceding sections we have discussed how quantum effects of the electrons are treated first of all within density-functional theory, but we emphasize that, from a conceptual point of view, treatments based on the Hartree-Fock approximation are fairly similar. There exist some few methods where the quantum treatment of the electrons is extended by a quantum treatment of (some of) the nuclei (see, e.g., refs. 60-62) but whereas the electrons still are treated within the standard electronic-strueture approaehes, the path-integral method of Feynman is used for the nuclei. The basie ideas behind these will be briefly outlined here followed by some few examples of their implieations. 

The path integral formulation of quantum theory provides a framework to describe the behavior of solvated electrons. Feynman used the approach to treat the slow moving electron in ionic crystals — the prototypical polaron problem. We have extended this theory, drawing on theories of the liquid state, to analyze the localization transition and related phenomena found with excess electrons in fluids. 

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