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arXiv:2112.09108 · American Journal of Physics

On the normalization and density of 1D scattering states

Chris L. Lin

Published Feb 21, 2024• 1 citation

Abstract

The normalization of scattering states is more than a rote step necessary to calculate expectation values. This normalization actually contains important information regarding the density of the scattering spectrum (along with useful details on the bound states). For many applications, this information is more useful than the wavefunctions themselves. In this paper, we show that this correspondence between scattering state normalization and the density of states is a consequence of the completeness relation, and we present formulas for calculating the density of states, which are applicable to certain potentials. We then apply these formulas to the delta function potential and the square well. We then illustrate how the density of states can be used to calculate the partition function for a system of two particles with a point-like (delta potential) interaction.

DOI: 10.1119/5.0082650

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The normalization of scattering states is more than a rote step necessary to calculate expectation values. This normalization actually contains important information regarding the density of the scattering spectrum (along with useful details on the bound states). For many applications, this information is more useful than the wavefunctions themselves. In this paper, we show that this correspondence between scattering state normalization and the density of states is a consequence of the completeness relation, and we present formulas for calculating the density of states, which are applicable to certain potentials. We then apply these formulas to the delta function potential and the square well. We then illustrate how the density of states can be used to calculate the partition function for a system of two particles with a point-like (delta potential) interaction.

The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.

The response of an arbitrary scattering problem to quasistatic perturbations in the scattering potential is naturally expressed in terms of a set of local partial densities of states and a set of sensitivities each associated with one element of the scattering matrix. We define the local partial densities of states and the sensitivities in terms of functional derivatives of the scattering matrix and discuss their relation to the Green's function. Certain combinations of the local partial densities of states represent the injectivity of a scattering channel into the system and the emissivity into a scattering channel. It is shown that the injectivities and emissivities are simply related to the absolute square of the scattering wave function. We discuss also the connection of the partial densities of states and the sensitivities to characteristic times. We apply these concepts to a \ensuremath{\delta} barrier and to the local Larmor clock. \textcopyright{} 1996 The American Physical Society.

In this paper it is considered a particle in an infinite potential well of special smooth form. To obtain eigenfunctions and eigenvalues of the problem, a method of Darboux transformations is applied. An orthonormal set of the eigenfunctions is presented. Some important properties of the eigenfunctions are shown. Principal characteristics of the particle in the well are discussed. A function of distribution of the probability density is analyzed. Conditions for determination of points of extrema of the probability density are formulated and corresponding transcendental equations are presented. An approximated method for determination of points of extrema is also proposed. Expected values of position and momentum of the particle are calculated explicitly. For some lower energy states, distributions of probability density are shown graphically in comparison with distributions for the infinite square well. As an additional useful result, some integrals of special combinations of trigonometric functions are presented.