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POINT INTERACTIONS IN QUANTUM MECHANICS

Chikhachev A.S.

Published May 24, 2022• 0 citations
Publisher

Abstract

Models of point interactions that approximate the real interactions of atomic particles in quantum mechanics are considered. In a one-point one-dimensional system, states are studied taking into account the eigenfield and a non-stationary problem is solved. The concept of a "point cluster" is introduced - a complex of charges that, at zero size, create the possibility of localization of a test particle in a region of finite size. An integral model of point interaction is proposed. States in three-dimensional two-point systems with "local isotropy" are studied.

DOI
10.5281/zenodo.6576677

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Models of point interactions that approximate the real interactions of atomic particles in quantum mechanics are considered. In a one-point one-dimensional system, states are studied taking into account the eigenfield and a non-stationary problem is solved. The concept of a "point cluster" is introduced - a complex of charges that, at zero size, create the possibility of localization of a test particle in a region of finite size. An integral model of point interaction is proposed. States in three-dimensional two-point systems with "local isotropy" are studied.
Models of the pointed interactions approximately describing real interactions of nuclear particles in quantum mechanics are considered. The concept of “a dot cluster”—a complex of charges which at the zero size create possibility of localization of a trial particle in the field of the final size is entered. States in one-dimensional systems, and also in three-dimensional systems with “a local isotropy” are studied. The conditions of dot systems characterized by the nonzero, including fractional, orbital moment were studied.
The present paper is devoted to the study of the few–body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play an important role in mathematical physics, since few body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [ 13 ]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the many body Hamiltonians can often be calculated using Bethe Ansatz [ 14 ]. In fact the well-known Yang-Baxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several particles [ 12 ] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt to construct a three body Hamiltonian describing three quantum particles in R 3 interacting through pairwise delta functional potentials is due to G.V. Skorniakov and K.A.Ter-Martirosian [ 29 ]. R.A.Minlos and L.D.Faddeev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in 278 the originally intended physical applications [ 24 , 25 ]. The operator was defined using the method of self–adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schödinger operators with delta potential in R 3 [ 10 ]. Almost three decades later semibounded three-body operators in dimension three with generalized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R 9 [ 17 , 18 , 19 , 27 , 30 ]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [ 7 , 26 ]. Different aspects of these models were analyzed recently [ 8 , 23 , 20 ]. A general approach to these operators is described in [ 7 , 22 ]. Let us mention that a realization of a many-body lower bounded Hamiltonian with point interactions for particles in R 3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [ 2 ]. The few-body systems of two dimensional particles with two-body interactions was studied in [ 11 ], where it was proven that the corresponding Hamiltonian is semibounded.
In previous papers, a generalized kinematic momentum was introduced in standard quantum mechanics, in order to address Einstein’s objection concerning the vanishing Bohmian velocity related to real stationary wave functions. The model was applied successfully to the one-dimensional case of a particle in a box and of a bouncing ball solving a nonlinear Schrödinger equation of the Ermakov–Pinney type. In this paper we develop the formalism giving a three-dimensional generalization of the model. In the stationary regime, the associated kinematic current is divergence free and can be locally represented as the curl of an auxiliary vector field C. We consider, as an application of the three-dimensional generalization, the case of the Hydrogen atom. Finally, a possible experimental test of the model is proposed by applying the formalism to an infinite box with an internal finite potential step. This example suggests a possible experimental setting in which the presence of a non-vanishing Ermakov–Pinney kinematic current could produce deviations from the standard predictions of the linear quantum theory.
Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.
We derive a variational cluster approximation for Heisenberg spin systems at finite temperature based on the ideas of the self-energy functional theory by Potthoff for fermionic and bosonic systems with local interactions. Partitioning the real system into a set of clusters, we find an analytical expression for the auxiliary free energy, depending on a set of variational parameters defined on the cluster, whose stationary points provide approximate solutions from which the thermodynamics of spin models can be obtained. We explicitly describe the technical details of how to evaluate the free energy for finite clusters and remark on specific problems and possible limitations of the method. To test the approximation we apply it to the antiferromagnetic spin chain and compare the results for varying cluster sizes and choices of variational parameters with the exact Bethe ansatz solution.