Diffusion through sharp potential barriers I. Exact solution

A. N. Malakhov

Ekaterina A. Volkova, A. M. Popov, Olga V. Tikhonova

The tunneling of a quantum system with an internal degree of freedom through a potential barrier is considered. Based on the exact numerical solution to the nonstationary Schrödinger equation, the tunneling of a model two-particle system through a potential barrier is studied and the dependences of the tunneling transparency of the barrier on the parameters of the wave packet that describes the system at the initial moment are obtained. A sharp increase in the tunneling probability related to the formation of a long-lived quasibound state of the system in the barrier region is demonstrated. A simple analytical model of the tunneling of a system with an internal degree of freedom that allows for a qualitative interpretation of the main features of the tunneling is constructed.

Donald J. Le Roy, Richard Roy, K. A. Quickert

Vadim Berdichevsky, Moshe Gitterman

Exact solutions are obtained for the diffusion in single- and square-well potentials, by using the Laplace transform method. We compare transmission of a classical particle through one barrier of given length and through two barriers with a well between them of the same overall length. It turns out that the existence of a well facilitates the transmission of a classical particle for intermediate times, which bears some resemblance to resonances tunneling through quantum barriers. \textcopyright{} 1996 The American Physical Society.

Domenico Oricchio

I use the Laplace transform to obtain the transmission coefficient for each number of potential barriers.There are a little number of close-form solution of Schröginger equation, and this is another exact solution.It is possible to obtain the tunnelling for electronic components (resonant-tunnelling diode), or neutronand proton diffusion in astrophysics (stellar nucleosyntheis) and nuclear physics (nuclear fission and fusion),or chemical tunnelling for two molecules reaction (that I solve in a generic case), or to obtain a quantumfilter for the energy spectrum (that I solve for some coating layers).