The coordinate matrix element of the time evolution operator, exp[−iĤt/ℏ], is determined by expanding (its exponent) in a power series in t. Recursion relations are obtained for the expansion coefficients which can be analytically evaluated for any number of degrees of freedom. Numerical application to the tunneling matrix element in a double well potential and to the reactive flux correlation function for a barrier potential show this approach to be a dramatic improvement over the standard short time approximation for the propagator. Its use in a Feynman path integral means that fewer ‘‘time slices’’ in the matrix product exp[(−i/ℏ)ΔtĤ]N, Δt=t/N, will be required. The first few terms in the present expansion constitute a fully quantum version of the short time propagator recently obtained by us using semiclassical methods [Chem. Phys. Lett. 151, 1 (1988)].
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator
exp(+sH), where
H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary
c-number integral of a
c-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian,
Heff, which is a function of the parameter
s and the
c-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function
ZN in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for
Heff depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying
exp(sHeff). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for
Heff. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
The exact calculation of the matrix elements is extraordinarily difficult beyond the second order in a perturbation expansion. Therefore, it is of crucial importance to examine the operator and vector series, with the purpose of gaining some insight into convergence of the pertinent physical matrix elements. A great advantage is that convergence of an operator expansion automatically guarantees the convergence of the corresponding series of vectors and matrix elements. The chapter examines in some detail a connection which exists between operator and vector expansions, thus establishing the quoted relation of the corresponding convergence radii. The importance of the convergence problem of the Neumann-Born expansions based upon the outlined concept of a comparative study of series of operators, vectors and matrix elements is in its application to the cases when the kernel K is not a compact operator.