Once again on quantum propagator for the step potential

Abstract

We present one another consideration of the propagator for a quantum particle in the presence of the step potential. This problem has been under treatment several times both as quantum and as stochastic ones but the explicit formulas derived in [12] for a problem in the "imaginary time representation" could hardly be used for computations in quantum case. Following mainly [5] the systematic exploration performed and the explicit expressions for the propagator in real time are constructed.

Solving the time-dependent Schrödinger equation via Laplace transform

“…What comes closer to the proposed method is the derivation of the propagator K(x, x ′ , t) from the Green function which means to invert the half-sided Fourier transform ( [16], [17])…”

“…We proceed with the potential step in section 4. The derivation of the exact solution in this case is significantly easier than the derivation of the 2 propagator in [18], [17] where the PDX method for path integrals was applied. Apart from being of similar simplicity as the most explicite solutions available in the literature ( [19], [17]) the obtained solutions admit a direct insight into the wave packet dynamics.…”

“…The derivation of the exact solution in this case is significantly easier than the derivation of the 2 propagator in [18], [17] where the PDX method for path integrals was applied. Apart from being of similar simplicity as the most explicite solutions available in the literature ( [19], [17]) the obtained solutions admit a direct insight into the wave packet dynamics. We show directly that sufficiently peaked wave packets with energy higher than the potential step will be slowed down as a corresponding classical particle which was already derived in [16] using the saddle point approximation.…”