Feynman path-integral formalism

In this section we shall expand upon the problem of one-dimensional motion in a potential V x). Although it is a textbook example, we use here the less traditional Feynman path-integral formalism, the advantage of which is a possibility of straightforward extension to many dimensions. In the following portion of the theory we shall use dimensionless units, in which h = i,k = 1 and the particle has unit mass. 

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. 

Straus et al. [223] noticed that the quantization of the librational and vibrational modes of water (by means of the Feynman path integral formalism [224]) can significantly influence the solvent free energy barrier and the thermodynamic driving force of the heterogeneous electron transfer process. For more information the reader is referred to the cited publications and the references therein. 

Action integrals, which are at the heart of the Feynman path-integral formalism, are employed as a formal tool in classical mechanics for generating Lagrange s equations of motion. A brief review of this postulate and its application is appropriate here and allows us to introduce some notation. Of course, the reader may consult any of a number of excellent texts for more detailed treatments of this subject [67-69]. 

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. 

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. 

Lee s discretization of nonrclativistic quantum mechanics is almost as straightforward as the discretization of classical mechanics discussed above he uses Feynman s path integral formalism [feyn65b]. 

Sese, L. M. 1994, Study of the Feynman-Hibbs effective potential against the path-integral formalism for Monte Carlo simulations of quantum many-body Lennard-Joncs systems . Mol. Phy.s. 81, 1297 1312. 

Other interesting treatments of the solid motion have been developed in which the motion of the solid s atoms is described by quantum mechanics [Billing and Cacciatore 1985, 1986]. This has been done for a harmonic solid in the context of treatment of the motion of the molecule by classical mechanics and use of a TDSCF formalism to couple the quantum and classical subsystems. The impetus for this approach is the fact that, if the entire solid is treated as a set of coupled harmonic oscillators, the quantum solution can be evaluated directly in an operator formalism. Then, the effect of solid atom motion can be incorporated as an added force on the gas molecule. Another advantage is the ability to treat the harmonic degrees of freedom of the solid and the harmonic electron -hole pair excitations on the same footing. The simplicity of such harmonic degrees of freedom can also be incorporated into the previously defined path-integral formalism in a simple manner to yield influence functionals (Feynman and Hibbs 1965). 

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. 

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) ... 

Feynman noted that the quantum mechanical centroid density, Pc(xc), can be defined for the path centroid variable which is the path integral over all paths having their centroids fixed at the point in space Xc. Specifically, the formal imaginary time path integral expression for the centroid density is given by... 

The thermal density matrix p plays a key role in Feynman s imaginary-time Path Integrals (PI) formalism and its application in Monte Carlo (MC) algorithms to compute physical properties of interest. In position space, it is given by [1-3] ... 

Feynman-Kleinert formalism Golden ratio Matsubara frequencies Matter s stability path integrals for chemistry Riemann s series Wiegner expansion... 

Such equivalence between the path integral Feynman-Kleinert formalism and the density functional Levy s one recommends the use of the Fe5mman-Kleinert density/densities for being implemented in density functionals with chemical relevance, the electronegativity for instance. 

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

---