One-Dimensional Problems in Quantum Mechanics

Michele Cini, Francesco Fucito, Mauro Sbragaglia

Shu Long

The one-dimensional finite-depth potential well problem under certain boundary conditions discussed in this paper is mainly the eigenvalue problem in quantum mechanics. The basis of this article is the solution of Schrodinger equation and the treatment of transcendental equation. In many quantum mechanics textbooks, the one-dimensional infinite potential well problem is regarded as the foundation of quantum mechanics because it is easy to handle and understand. On this basis, this paper discusses the more challenging finite-depth potential well situation. In this paper, the transcendental equation is obtained by analogy between the wave function with time and the continuity of the boundary conditions. In order to solve the problem that transcendental equation is difficult to solve in this process, this paper adopts the method of solving numerical solutions by using images, which includes two different cases of odd and even functions. This paper reveals the possibility of particles appearing outside the well in this problem, and on this basis, some discussions and analysis are made on the one-dimensional asymmetric finite-depth potential well problem.

Won Sang Chung, HASSAN HASSANABADI

In this paper, we consider the quantum mechanics with Dunkl derivative. We use the Dunkl derivative to obtain the coordinate representation of the momentum operator and Hamiltonian. We introduce the scalar product to find that the momentum is Hermitian under this inner product. We study the one-dimensional box problem (the spin-less particle with mass m confined to the one-dimensional infinite wall). Finally, we discuss the harmonic oscillator problem.

Emilio d’Emilio, Luigi E. Picasso

Vladimir K. Ignatovich

An elegant mathematical method is demonstrated with the help of simplest one-dimensional problems of quantum mechanics. This method is then applied to calculation of scattering on one-dimensional periodic systems. Generalization of the method for calculation of scattering in three dimensional periodic media and for spinor particles is pointed out.