Toward a new time-dependent path integral formalism based on restricted quantum propagators for physically realizable systems

Abstract

Various mathematical properties of the exact free and full short-real-time propagators in the coordinate representation are discussed. The viewpoint is emphasized that these properties, which prevent the use of the exact path integral formalism as a basis for straightforward computations and result in bizarre behavior wholly at odds with our (macr~pically conditioned) intuition, are neuer relevant, in connection with any experimentally realizable system. These computationally disasterous mathematical properties of the kinetic energy part of the propagator may be removed by means of restricting the wave packets which will be propagated to L2 functions which can be fitted, sufficiently accurately, by a polynomial of a specific degree within the envelope of the fitting function. This is an example of 'prefiltering" oscillations whose frequency increases without bound (i.e., as the distance Ix'-XI travelled in short time T ) . The decay in the kinetic energy, which characterizes any physically realizable system, has been achieved using our continuum distributed approximating functions, which limit the class of wave packets which can be propagated. We also discuss the role of the potential energy. Again, difficulties arise which could be addressed by potential energy cutoff procedures. However, the more desireable possibility of developing a smoothly prefiltered full propagator is pointed out.

Particular and homogeneous solutions of time-independent wavepacket Schrödinger equations: calculations using a subset of eigenstates of undamped or damped Hamiltonians

“…The expressions in Eqs. (24) and (25) (or, alternatively, Eqs. 34,35), however, are functions of 1=E ÀẼ k , which is singular at E Ẽ k .…”

Symmetry-adapted distributed approximating functionals: Theory and application to the ro-vibrational states of H3+

“…where the accuracy is governed by the Fourier-space characteristics of the DAF window. [8][9][10][11][12][13] One may approximate the integral over xЈ by an appropriate sampling to obtain a simple and accurate [8][9][10][11][12][13] way to calculate the value of a function and its k-th derivative, at any point on or off a grid chosen for discretization, as a matrix-vector product. We shall employ the Hermite-DAF, [8][9][10][11][12][13] ␦ DAF…”

“…Some of the popular choices for bases include the Fourier [1][2][3][4] functions, eigenstates for various bound degrees of freedom, 5 and the discrete variable representation ͑DVR͒. 6,7 In recent years, a new approach, [8][9][10][11][12][13] based on distributed approximating functionals ͑DAF͒, has been introduced as a means of representing any derivative operator accurately. It has been used to obtain suitable coordinate representations for both the kinetic energy operator and the free-propagator.…”

Uniform Approximation of Functions with Discrete Approximation Functionals

“…They constructed the operators D n so that the kernel matrix (U { D n )(r$2, r2) is strongly banded. Numerical calculations employing the operators D n (which Hoffman et al called distributed approximation functionals) proved to be efficient, accurate, and robust [5,9,12].…”