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arXiv:1709.05783 · Europhysics Letters (EPL)

Exact solution for many-body Hamiltonian of interacting particles with linear spectrum

M. V. Entin, Leonid Braginsky

Published Oct 1, 2017• 5 citations
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Abstract

The exact solution of the Schrödinger equation for the one-dimensional system of interacting particles with linear dispersion law in an arbitrary external field is found. The solution is reduced to two groups of particles moving with constant velocities in the opposite directions with a fixed distance between the particles in each group. The problem is applied to the edge states of the 2D topological insulator.

DOI
10.1209/0295-5075/120/17003

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The exact solution of the Schrödinger equation for the one-dimensional system of interacting particles with linear dispersion law in an arbitrary external field is found. The solution is reduced to two groups of particles moving with constant velocities in the opposite directions with a fixed distance between the particles in each group. The problem is applied to the edge states of the 2D topological insulator.
Continuation of previous work (ibid., vol.11, no.7, p.1227 (1978)). Earlier exact solutions of Schrodinger's equation for translation-invariant systems of particles interacting by Hooke's law pair potentials are augmented to include systems consisting of an arbitrary number S of groups of identical particles. Exact solutions can always be found whenever the number of distinct masses plus the number of distinct coupling constants does not exceed (2S+1). The case S=2 is solved in detail and is applied to show that harmonic matter is never stable.
Exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schrödinger equation is applied for a solution of a two--dimensional (2D) problem for two particles interacting via Kratzer, and modified Kratzer potentials. We found the exact bound state solutions of the two--dimensional Schrödinger equation with Kratzer--type potentials and present analytical expressions for the eigenvalues and eigenfunctions. The eigenfunctions are given in terms of the associated Laguerre polynomials.
A method of solving the Schrödinger equation based on the use of constant particle-particle interaction potential surfaces (IPS) is proposed. The many-body wave function is presented in a configuration interaction form, with coefficients depending on the total interaction potential. The corresponding set of linear ordinary differential equations for the coefficients was developed. To reduce the computational work, a hierarchy of approximations based on interaction potential surfaces of a part of the particle system was worked out. The solution of a simple exactly solvable model and He-like ions proves that this method is more accurate than the conventional configuration interaction method and demonstrates a better convergence with a basis set increase.
A new approach to the problem of description of the stationary motion of a quantum particle in the field of a one-dimensional arbitrary potential is developed. It is shown that the wave function of infinite motion, with the accuracy to two arbitrary constants, can be expressed by means of an arbitrary single solution for some set of linear differential equations of the first order. It is shown that one general property of the Schrödinger equation solutions lies in the basis of many known methods of the problem consideration, such as the method of integral equations, transfer-matrix method, imbedding method, and method of combination of scattering parameters. Within the framework of the proposed approach, the connection between the above-mentioned methods becomes more transparent.