Numerical Calculations of Effective Potentials in One-dimensional Quantum Mechanics

S. S. Kancharla

W. Polak

This article presents a simple, yet general and exact analytic-numerical method of solving one-dimensional quantum mechanics problems. It is based on the concept of dividing an arbitrary potential into convenient segments and analyzing the multiple reflection of a plane matter wave between neighboring potential walls. Amplitude reflection and transmission coefficients of each segment are the basis for the analysis. A simple iterative technique for calculating them by a square barrier approximation is given. This method is applicable to various potential barriers, wells, and periodic structures including continuous variations of potential energy and particle effective mass.

Zbigniew Ambrozinski

The tunneling effect is the most popular phenomenon of quantum physics and is present in modern physical theories. Still, the most important features of this effect are already present in toy models - low dimensional quantum mechanics with high potential barriers. Even these simple systems cannot be solved analytically and approximations have to be made. Tunneling is closely related to splitting of the ground state energy which is degenerate in the classical limit. In this thesis we use two methods to estimate the energy splitting in one dimensional quantum mechanics. In semi-classical approximation the splitting is calculated in the weak coupling limit. The other approach, Tamm-Dancoff method gives very precise numerical results for finite coupling. Comparing these two approaches we are able to confirm results of the semi-classical approximation and determine range of its applicability. Several potentials are considered: anharmonic double well, cosine potential and cosine potential in periodic space with finite number of minima. We also address a non-generic problem of anharmonic triple well potential.

Tôru Sakai

和夫 斉藤

Hassanabadi, H., Sargolzaeipor, S., Chung, W.S.

In this paper, after introducing superstatistics in quantum mechanics, first, the Schrödinger equation in one-dimensional in the presence of q-deformed Morse potential is derived. We use an analytical methodology to calculate the energy eigenvalues. Then the thermodynamic properties of the system are calculated. Furthermore, we report effects of all parameters in the problems on the thermodynamic quantities.