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Multiscale and Multiphysics Mechanics

A split kinetic energy solution scheme applied to various delta potentials in quantum mechanical systems

Yu-Hsin Chen, Sheng Der Chao

Published Jul 25, 2016• 2 citations
Publisher

Abstract

In this work, we extend the split previously developed kinetic energy (dubbed as KEP) method Mineo and Chao (2012) by modifying the mass parameter to include the mass. We first show how to separate the total system into the subsystems with 3 attractive delta potentials by using the KEP method. For repulsive delta potentials, we introduce negative mass terms. Two cases are demonstrated using the negative mass terms for repulsive delta potential problems in quantum mechanics. Our work shows that the KEP solution scheme can be used to obtain not only the exact energies but also the exact wavefuctions very precisely.

DOI
10.12989/mmm.2016.1.3.189

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In this work, we extend the split previously developed kinetic energy (dubbed as KEP) method Mineo and Chao (2012) by modifying the mass parameter to include the mass. We first show how to separate the total system into the subsystems with 3 attractive delta potentials by using the KEP method. For repulsive delta potentials, we introduce negative mass terms. Two cases are demonstrated using the negative mass terms for repulsive delta potential problems in quantum mechanics. Our work shows that the KEP solution scheme can be used to obtain not only the exact energies but also the exact wavefuctions very precisely.
In this work, we extend the previously developed split kinetic energy (dubbed KEP) method by Mineo and Chao (2012) by modifying the mass parameter to include the negative mass. We first show how to separate the total system into the subsystems with 3 attractive delta potentials by using the KEP method. For repulsive delta potentials, we introduce "negative" mass terms. Two cases are demonstrated using the "negative" mass terms for repulsive delta potential problems in quantum mechanics. Our work shows that the KEP solution scheme can be used to obtain not only the exact energies but also the exact wavefunctions very precisely.
We extend a method previously presented for the three-body problem with attractive interparticle potentials to the case of repulsive interparticle potentials, with periodic boundary conditions on each particle. As before, we decompose the wave function into three parts, and from the Schrödinger equation write equations for these parts. We observe that, in one-dimension and for δ-function potentials,these equations can be easily solved numerically, and we present these solutions for potential“strengths” ranging from zero to infinity. We then discuss a more general method for solving these equations, which method involves expansions in a certain set of two-body functions. This general method is not necessarily limited to either one dimension or δ-function potentials, but, as a check on it, we do apply it to that case and get good agreement with the previous numerical results. As a further exploration of the method, we apply it to square well potentials in one dimension. In an Appendix we discuss the set of two-body functions that we use.
Based on the idea of adiabatic symmetry, we present a novel basis set expansion method—the kinetic energy partition (KEP) method—for solving quantum eigenvalue problems. Broken symmetry is responsible for quantum entanglement in many-body systems via parametric non-adiabatic corrections. Starting from simple one-particle-in-one-dimension problems, we gradually increase the complexity in the number of particles and the interaction patterns. Our goal in the mini-review is to advocate for the utility of the KEP method in front-line research, in particular for research beginners in quantum many-body problems.
Studies the ground-state energy of a system of N identical bosons, each having mass m, which interact in one dimension via the pair potential V(x)= gamma f(x/a) and obey non-relativistic quantum mechanics. A recent energy lower bound based on the known exact solution for the delta potential is compared to an earlier bound provided by the equivalent two-body method. The delta bound is good whenever the potential is very narrow and deep. For any bounded potential, the earlier energy bound is always better for large N. Detailed results and graphs are given for the sech-squared potential and the square-well potential.
We study perturbation theory in certain quantum mechanics problems in one dimension in which the perturbing potential diverges at some points, even though the energy eigenvalues are smooth functions of the coefficient of the potential. We discuss some of the unusual techniques which are required to obtain perturbative expansions of the energies in such cases. These include a point-splitting prescription with a δ′′ function for expanding around the Dirichlet (fermionic) limit of the δ function potential, and performing a similarity transformation to a non-Hermitian potential in the Calogero–Sutherland model. As an application of the first technique, we study the ground state of the δ function Bose gas near the fermionic limit.
By splitting a Hamiltonian into two parts, using the solvability of eigenvalue problem of one part of the Hamiltonian, proving a useful identity and deducing an expansion formula of power of operator binomials, we obtain an explicit and general form of time evolution operator in the representation of solvable part of the Hamiltonian. Further we find out an exact solution of Schrödinger equation in a general time-independent quantum system, and write down its concrete form when the solvable part of this Hamiltonian is taken as the kinetic energy term. Comparing our exact solution with the usual perturbation theory makes some features and significance of our solution clear. Moreover, through deriving out the improved forms of the zeroth, first, second and third order perturbed solutions including the partial contributions from the higher order even all order approximations, we obtain the improved transition probability. In special, we propose the revised Fermi's golden rule. Then we apply our scheme to obtain the improved forms of perturbed energy and perturbed state. In addition, we study an easy understanding example to illustrate our scheme and show its advantage. All of this implies the physical reasons and evidences why our exact solution and perturbative scheme are formally explicit, actually calculable, operationally efficient, conclusively more accurate. Therefore our exact solution and perturbative scheme can be thought of theoretical developments of quantum dynamics. Further applications of our results in quantum theory can be expected.