Feynman’s path integral formulation

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. 

The quantum mechanical free energy barrier, A.g, can be evaluated by Feynman s path integral formulation [59], where each classical coordinate is replaced by a ring of quasiparticles that are subjected to the effective quantum mechanical potential... 

S. M. Blinder is Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor. Born in New York City, he completed his PhD in Chemical Physics from Harvard in 1958 under the direction of W. E. Moffitt and J. H. Van Vleck (Nobel Laureate in Physics, 1977). Professor Blinder has over 100 research publications in several areas of theoretical chemistry and mathematical physics. He was the first to derive the exact Coulomb (hydrogen atom) propagator in Feynman s path-integral formulation of quantum mechanics. He is the author of three earlier books Advanced Physical Chemistry (Macmillan, 1969), Foundations of Quantum Dynamics (Academic Press, 1974), md Introduction to Quantum Mechanics in Chemistry, Materials Science and Biology (Elsevier, 2004). 

This example of a flexible polymer chain in the presence of an external field serves to demonstrate how the use of the limit of a continuous chain leads to distributions that satisfy differential equations. This limit therefore introduces an analogy between the configurations of the continuous equivalent chain and the path of a particle which is undergoing Brownian or diffusive motion. - It is also useful to relate this continuous chain description to Feynman s path integral formulation of quantum me-chanics. This relationship permits the exploitation of mathematical analogies which might lead to suitable approximation schemes. Reference is made to the book by Feynman and Hibbs for certain mathematical details which can be transcribed between quantum mechanical and polymer problems. 

Typically, it is now possible in molecular quantum dynamics to treat systems with an ab initio potential energy surface and curvilinear coordinates up to 6-7 atoms in full dimensionality such as (H2)3 [152], CH [102], and HsO [153,154]. When a model is used for the potential and/or a description with normal coordinates is adopted, it is possible to deal with systems a little bit larger such as pyrazine, including two electronic states coupled through a conical intersection [155] or malonaldehyde [156]. One can also mention the H - - CH4 system that undergoes a reactive process [157]. We will see later that the recent Multi-Layer MCTDH approach will allow one to treat even larger systems. Note also that Feynman s path integral formulations have been applied to study condensed matter [158], but they will not be addressed in the present book. 

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... 

Eq. (1) represents the stationary phase approximation to the full quantum mechanical propagator, which in Feynman s path integral formulation takes the form of a sum over all paths. This can be seen directly by realizing that the stationary phase condition amounts precisely to Hamilton s principle, and inclusion of quantum fluctuations within a tube of size fi around classical trajectories gives rise to the semiciassical prefactor. 

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. 

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