Contracted distributed approximating functions: Derivation of non-oscillatory free particle and harmonic propagators for Feynman path integration in real time

Abstract

Contracted continuous distributed approximating functions (CCDAFs) have been developed. In particular, it has been shown that, continuous distributed approximating functions (CDAFs) based on standard orthogonal polynomials can be contracted to functions formed as the product of a weight function and the sinc function or a Bessel function of the first kind. The CCDAFs of Hermite type have been applied to derive new expressions for the coordinate representation of the free particle evolution operator and that of the evolution operator of harmonic oscillator. These new expressions of free particle and harmonic propagators have as compact mathematical form as Makri’s effective free propagator [N. Makri, Chem. Phys. Lett. 159, 489 (1989)] and Gaussian decay identical to that of the CDAF class free and harmonic propagators due to Kouri et al. [D. J. Kouri, W. Zhu, X. Ma, B. M. Pettitt, and D. K. Hoffman, J. Phys. Chem. 96, 9622 (1992)] and Marchioro et al. [T. L. Marchioro II, M. Arnold, D. K. Hoffman, W. Zhu, Y. Huang, and D. J. Kouri, Phys. Rev. E50, 2320 (1994)], respectively. The Gaussian decay of a CCDAF Hermite free propagator has been shown to be the result of including momentum eigenstates in the propagator which have momenta larger than the momentum of the wave packet of largest momentum that still can be well approximated by the CCDAF considered.