Feynman path

The quantity is the Feynman path integral centroid density [43] that is understood to be expressed asymptotically as... 

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

Cao, J., Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties. J. Chem. Phys. 100 (1994) 5093-5105 II Dynamical properties. J. Chem. Phys. 100 (1994) 5106-5117 III. Phase space formalism and nalysis of centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6157-6167 IV. Algorithms for centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6168-6183 V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys. 101 (1994) 6184 6192. 

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. 

Another example of slight conceptual inaccuracy is given by the Wigner function(12) and Feynman path integral(13). Both are useful ways to look at the wave function. However, because of the prominence of classical particles in these concepts, they suggest the view that QM is a variant of statistical mechanics and that it is a theory built on top of NM. This is unfortunate, since one wants to convey the notion that NM can be recovered as an integral part of QM pertaining to for macroscopic systems. 

The wave function is an irreducible entity completely defined by the Schrbdinger equation and this should be the cote of the message conveyed to students. It is not useful to introduce any hidden variables, not even Feynman paths. The wave function is an element of a well defined state space, which is neither a classical particle, nor a classical field. Its nature is fully and accurately defined by studying how it evolves and interacts and this is the only way that it can be completely and correctly understood. The evolution and interaction is accurately described by the Schrbdinger equation or the Heisenberg equation or the Feynman propagator or any other representation of the dynamical equation. 

Albeverio, S.A. and Hpegh-Krohn, R.J., Mathematical Theory of Feynman Path Integrals, Lecture Notes in Mathematics 523, Springer, Berlin, Heidelberg, and New York, 1976. 

To explain the effect of the GP on the nuclear dynamics [i.e., to explain the difference between the dynamics described by an encircling T g(< )) and an encircling T n(< ))], we need to compare the topology of T g(< )) with the topology of T n(< ))- In Section II. C, we review how this can be done using the homotopy of the Feynman paths [41-45] that make up these wave functions. But first, to demonstrate the simplicity of the problem, we use the diagrammatic approach developed in Refs. [27 and 28]. 

Figure 5. Examples of Feynman paths belonging to different homotopy classes, illustrating how the winding number n is defined.

The separation of the Feynman paths in Eq. (10) is equivalent to the splitting of the wave function into and in Eq- (6). To demonstrate this, we connect the Kernel to the wave function using [48],... 

Figure 6. Diagram showing how the winding number n of the Feynman paths should be defined with respect to the cut line. In (a), the cut line (chains) is placed between (() = — and 2n — in (b), between (() = ti/4 and —In/A. In (c), the wave function describes a unimolecular reaction, in which the initial state occupies the (gray shaded) area shown. Feynman paths originate from all points within this area (inset) their winding number n is defined with respect to the common cut line.

In general, it is difficult to map contributions from different reaction paths onto the DCS. However, Eq. (6) teUs us that, in a reaction with a Cl, one can easily map the contributions from the e and o (Feynman) paths onto the DCS. Since Eq. (6) applies to the entire wave function, we can apply it to the asymptotic limit of the wave function in Eq. (14), and thus to and to obtain... 

We can represent this function in the single space, provided we use a common cut line for all three components. This is shown schematically in Fig. 17. Use of the common cut line is equivalent to taking the linear combinations in the double space, then cutting a 27i-wide section out of the entire (4)). The winding numbers n of the Feynman paths that enter the three equivalent reagent channels must all be dehned with respect to the common cut line, since they are analogous to paths starting at different points in the initial state of a unimolecular reaction (Section 11.D). 

Figure 18. Complete unwinding of an encircling nuclear wave function >0 by mapping onto higher cover spaces, (a) The function in the single space (b) e in the double space (c) 4 in the quadruple space (d) schematic picture of in a 2hn cover space. In each case, will be completely unwound if it contains contributions from Feynman paths belonging to (b) 2, (c) 4, and (d) h different winding-number classes.

When higher n Feynman paths contribute to the wave function, one has simply to apply repeatedly the single- to double-space mapping, until the nuclear wave function is completely unwound (in the sense just defined). Thus, if the wave function contains only n = —2, —1,0,1 paths, then we need to compute a function in the double space that satisfies the boundary condition 4 (([)) = —(cj) + 4ti). Adding this function to the [which satisfies he(( )) = he(( ) + 4ti)] then gives a new function, [ 4(( )), which occupies the quadruple space (j) = 0 8ti (see Fig. 18c). This new quadruple-space wave function will be completely unwound, such that there is a gap between its clockwise and counterclockwise branches. The n= 2, 1,0,1 contributions will lie in the... 

Clearly, the above procedure can be continued (in principle) as many times as required. Thus, if the wave function includes n = —4 3 paths, we have simply to dehne the function I 4((t)) = —+ 8ti), and then map onto the (j) = 0 16ti cover space, which will unwind the function completely. In general, if there are h homotopy classes of Feynman paths that contribute to the Kernel, then one can unwind ihG by computing the unsymmetrised wave function ih in the 0 2hn cover space. The symmetry group of the latter will be a direct product of the symmetry group in the single space and the group... 

To clarify, the complete unwinding of the wave function is not required to explain the effect of the GP. The latter affects only the sign of the odd n Feynman paths with respect to the even n paths, and is thus explained completely once one has unwound these two classes of path by mapping onto the double space. The complete unwinding explains the interference within the even n and odd n contributions, by unwinding each of them further, into the contributions from individual values of n. 

Of course, the distinction between reactive- and bound-state wave functions becomes blurred when one considers very long-lived reactive resonances, of the sort considered in Section IV.B, which contain Feynman paths that loop many times around the CL Such a resonance, which will have a very narrow energy width, will behave almost like a bound-state wave function when mapped onto the double space, since e will be almost equal to Fo - The effect of the GP boundary condition would be therefore simply to shift the energies and permitted nodal structures of the resonances, as in a bound-state function. For short-lived resonances, however, Te and To will differ, since they will describe the different decay dynamics produced by the even and odd n Feynman paths separating them will therefore reveal how this dynamics is changed by the GP. The same is true for resonances which are long lived, but which are trapped in a region of space that does not encircle the Cl, so that the decay dynamics involves just a few Feynman loops around the CL... 

Feynman paths that loop a different number of times around the Cl are decoupled from one another in the sum-over-paths, with the result that the nuclear wave function can be split into separate contributions from the even- and odd-looping paths. The sole effect of the GP is to change the relative sign of these two components. 

Hence, the GP has a much milder effect on reactive systems than on bound-state systems. This difference has been overlooked in the past, but becomes apparent on noting that an encircling bound-state function contains Feynman paths that loop an infinite number of times around the Cl [28]. Consequently, the encirclement of a bound-state wave function is much stronger than that of a reactive wave function the bound wave function cannot be unwound from around the Cl, whereas the reactive wave function can. One consequence of this is that the separation into even- and odd-looping paths yields no information about the dynamics of a bound state system, in which these two contributions are necessarily equal and opposite [28]. 

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. 

---