Showing papers on "Finite potential well published in 1982"

Positive energy Weinberg states for the solution of scattering problems

Abstract: Positive energy Weinberg states are defined and numerically calculated in the presence of a general complex Woods-Saxon potential. The numerical procedure is checked for the limit of a square well potential for which the Weinberg states and the corresponding eigenvalues are known. A finite number $M$ of these (auxiliary) positive energy Weinberg states are then used as a set of basis functions in order to provide a separable approximation of rank $M$, ${V}_{M}$, to a potential $V$, and also to the scattering matrix element $S$ which obtains as a result of the presence of $V$,${S}_{M}$. Both ${V}_{M}$ and ${S}_{M}$ are obtained by means of algebraic manipulations which involve the matrix elements of $V$ calculated in terms of the auxiliary positive energy Weinberg states. Next, expressions are derived which enable one to iteratively correct for the error in $V\ensuremath{-}{V}_{M}$. These expressions are a modified version of the quasiparticle method of Weinberg. The convergence of ${S}_{M}$ to $S$, as well as the first order iteration of the error in ${S}_{M}$, is examined as a function of $M$ for a numerical example which uses a complex Woods-Saxon potential for $V$ and assumes zero angular momentum. With $M=5$ and one iteration an error of less than 10% in $S$ is achieved; for $M=8$ the error is less than 1%. The method is expected to be useful for the solution of large systems of coupled equations by matrix techniques or when a part of the potential is nonlocal.NUCLEAR REACTIONS Scattering theory, expansions on a basis set of positive energy Weinberg states, removal of truncation error by the quasiparticle method. Normal mode effective scattering channels.

Continuum eigenfunction expansions and resonances: A simple model

Abstract: A one‐dimensional square‐well model of an atom in an electric field is solved exactly. This problem is instructive for two reasons: (1) The Hamiltonian has a continuous spectrum, but the eigenfunctions do not reduce asymptotically to plane waves. Therefore the general Titchmarsh–Kodaira theory of eigenfunction expansions is needed to normalize the eigenfunctions properly. The normalization is expressed by a spectral density function, ρ(E). (2) ρ(E) exhibits ’’bumps’’ at values of the energy at which the electron wave function’s amplitude inside the well is particularly large. These are resonant states of the system. The gradual sharpening of the resonances into discrete bound states as the electric field is turned off is demonstrated.

Resonances and analyticity of scattering wave function for square‐well‐type potentials

Abstract: In this paper we extend our previous analysis of the scattering of wave packets in one dimension to the case of the square‐well potential. The analytic properties of the general scattering solution are emphasized thereby making the analysis useful as introductory material for a more sophisticated S‐matrix treatment. The square‐well model is particularly interesting because of its application to the deuteron problem. Resonance scattering, barrier penetration, time delay, and line shape are discussed at the level of the first‐year graduate student.

Quantum solutions for a symmetric double square well

Abstract: The quantum behavior of a wave packet in a one‐dimensional infinite square well with a finite barrier in the center is considered. Computer generated plots are presented that lead to useful analytic approximations for finding eigenvalues of the Schrodinger equation and for explaining the time dependence of wave packets.

One‐dimensional hydrogen atom in an infinite square well

Abstract: The effect of boundary conditions on solutions to the Schrodinger Equation is demonstrated.

Discontinuity in the first derivative of the Schrödinger wave function

Abstract: Bound state problems in quantum mechanics are considered for the edge of a square well potential and singular points of a Coulomb and a δ‐function potential. (AIP)

Simple calculation of the level splitting for the double well potential

Abstract: It is shown that for the double-well potential three similarly constructed pairs of solutions of the Schr\"odinger equation can be derived which are such that they can be linked in regions of common validity. Evaluating the boundary conditions satisfied by the even and odd wave functions at a minimum, several analogous transcendental equations are obtained from which the splitting of (asymptotically degenerate) eigenvalues can be deduced.

Semiclassical statistical mechanics of a two‐dimensional fluid

Abstract: Expansions are obtained for the radial distribution function and free energy for a two‐dimensional hard‐core fluid in the semiclassical limit, using the ‘‘modified’’ Wigner–Kirkwood expansion method. These results are used to obtain expressions for the density‐independent part of the radial distribution function and the first‐order density correction to it. Quantum corrections to the second and third virial coefficients are discussed in detail. Explicit results are given for the Sutherland, Yukawa‐tail, Wood–Saxan, square‐well, and Lennard‐Jones (12‐6) pair potential models.

Quantum quadratic-attractive plus quartic repulsive potential in a box

Abstract: The solution of the Schrodinger equation for the two well oscillator in a symmetric box is formulated exactly, and high-accuracy numerical results are obtained for the lowest states. Perturbative solutions for boxes whose walls are (i) fairly close to each other, (ii) in the vicinity of the inflection points of the potential, (iii) at the position of the minima of the potential, and (iv) very far from each other are also obtained and compared with the exact ones.

Excitation spectrum and structure factor for interacting bosons

Abstract: Lennard-Jones, Gaussian and square well potentials have been used to obtain Ek and the structure factor S(k) using the reaction matrix formalism. The results obtained for Ek and S(k) for different potentials have been compared with each other and also with the experimental results. It is found that the Gaussian potential gives better results. The structure factor S(k) has been calculated using Berdahl's relation for S(k). The shape of S(k) differs markedly from the one obtained using a polynomial expression for S(k). However, the value obtained for the velocity of sound using a square well potential agrees very well with the experimental value.