Path integral techniques quantum mechanics

The path integral technique was first proposed by Feynmann (Feynmann Hibbs, 1965). The purpose of this technique was to deal with questions in quantum mechanics. It has been applied to the study of the statistical mechanics of polymer systems (Kreed, 1972 Doi Edwards, 1986) and liquid crystalline polymers as well (Jahnig, 1981 Warner et al, 1985 Wang Warner, 1986). The path integrals relate the configurations of a polymer chain to the paths of a particle when the particle is undergoing Brownian or diffusive motion. 

Finally, we note that quantum mechanical tunneling effects may also be incorporated through the use of path integral techniques [14]. These effects are believed to be important in electron transfer processes in condensed phases and proteins [15-16]. 

In this section we look briefly at the problem of including quantum mechanical effects in computer simulations. We shall only examine tire simplest technique, which exploits an isomorphism between a quantum system of atoms and a classical system of ring polymers, each of which represents a path integral of the kind discussed in [193]. For more details on work in this area, see [22, 194] and particularly [195, 196, 197]. 

The PIQMC method is the result of coupling of Feynman s path integral formulation of quantum mechanics with Monte Carlo sampling techniques to produce a method for finite temperature quantum systems. The calculations are not much more complicated than DQMC and produce a sum over all possible states occupied as for a Boltzmann distribution. In the limit of zero temperature... 

The path integral formulation of quantum statistical mechanics has become a powerful technique for describing quantum effects in liquids. " Path integrals introduce an effective quantum correction to the classical interaction potential. The classical two-body potential between two atoms of mass ntj and with coordinates x, and Xy is replaced by an effective potential between two ring polymers, each held together by a harmonic nearest-neighbor interaction,... 

The present article presents an introduction to the path integral formulation of quantum dynamics and quantum statistical mechanics along with numerical procedures useful in these areas and in electronic structure theory. Section 2 describes the path integral formulation of the quantum mechanical propagator and its relation to the more conventional Schrddinger description. That section also derives the classical limit and discusses the connection with equilibrium properties in the canonical ensemble, Numerical techniques are described in Section 3. Selective chemical applications of the path integral approach are presented in Section 4 and Section 5 concludes. 

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