Exact space-time propagator for the step potential

Abstract

We devise and apply a method for the calculation of the exact propagator for the Schrodinger equation for the one-dimensional Hamiltonian &(p,x)=p /2m + Voe(x). The limiting cases Vo -+0 (free particle) and Vo~~(opaque wall) are checked. The method is also applicable to any piecewiseconstant potential.

Path-integral analysis of arrival times with a complex potential

“…(12) therefore essentially reduces to the easier problem of calculating it between two points lying on x = 0. This can sometimes be evaluated by a mode sum calculation [13], even though the full propagator Eq. (12) is not necessarily calculable in this way.…”

The propagator for step potential and mass using path decomposition method

“…Problems with a step potential and a rectangular barrier potential in [11,12] and [13] were exactly resolved with the spectral summation method [15]. In [3,10,13,16,17] and [18], the calculation by the path integral approach of the Green's function with piecewise flat potentials constituted a good generalizing step. A new interesting device in path integrals is the path decomposition method PDX which was invented by Aurbach and Kivelson [19].…”

“…The basic idea of this technique is to decompose a propagator by grouping together the paths in the Feynman sum that cross a given surface (or a point in one dimension). This tool was successfully used in [20,21] to compute the traversal time distribution in quantum tunneling for the case of the square barrier, by Carvalho [10] and Yearsley [18] to study the step potential model, and to prove the method of images by Schulman [22]. In the present work we study a particle with coordinate dependent mass (step mass) and moving in one dimension under a step potential action.…”

“…3 and App A. For the coherence of our result with standard expression of the step potential propagator with constant mass [10] the limiting case is established in App. B. Sec.…”

“…In [6] the background field method was the tool for computing perturbatively in powers of the Planck constant the effective action for a quantum particle with coordinate dependent mass. The potential and the effective mass were often modeled by many authors using piecewise constant functions such that their solutions could realized exactly in different approaches [3,[10][11][12] and [13]. The Feynman path integral formalism [14] is adequate in the quantification of many problems in physics; whereas potentials with first order discontinuities remained hard to tackle within this framework.…”