The Four-Well Potential

Y. P. Varshni

The relative convergences of the WKB and SWKB approximations are examined by calculating the eigenenergies for four potentials by one- and two-term WKB and SWKB approximations. Exact eigenenergies for these four potentials are also calculated by a numerical integration of the Schrodinger equation. Varied results are found for the four potentials. It is found that in general the effect of the second term in the WKB and SWKB approximations depends on the potential, the parameters involved and the quantum number of the state. No simple generalizations are possible.

Michel Rousseau, J. P. Mathieu

Abstract: In this work, we present a detailed study of a one-dimensional Schrödinger equation in the presence of quantum Gaussian well interaction. Further, we investigate the approximate solutions by using the harmonic oscillator approximation, variational principle, four-parameter potential fitting and numerical solution using the finite-difference method. The parabolic approximation yields an excellent energy value compared with the numerical solution of the Gaussian system only for the ground state, while for the excited states, it provides a higher approximation. Also, the analytical bound-state energies of the four-parameter potential under the framework of the Nikiforov-Uvarov (NU) method have been used after getting the suitable values of the potential parameters using numerical fitting. The present results of the system states are found to be in high agreement with the well-known numerical results of the Gaussian potential. Keywords: Gaussian potential, One-dimensional Schrödinger equation, Nikiforov- Uvarov (NU) method, Four-parameter potential. PACS: 03.65.−w; 02.90.+p; 12.39.Pn.

Francisco Antônio Bezerra Coutinho, Yukihisa Nogami, Lauro Tomio

We consider a four-parameter family of point interactions in one dimension. This family is a generalization of the usual -function potential. We examine a system consisting of many particles of equal masses that are interacting pairwise through such a generalized point interaction. We follow McGuire who obtained exact solutions for the system when the interaction is the -function potential. We find exact bound states with the four-parameter family. For the scattering problem, however, we have not been so successful. This is because, as we point out, the condition of no diffraction that is crucial in McGuire's method is not satisfied except when the four-parameter family is essentially reduced to the -function potential.

Richard H. Enns, George C. McGuire