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Communications in Theoretical Physics

A Particle Interacting with V-shaped Potential Decorated by a Dirac Delta Function Interaction at Center

Wang Xin, Tang Liang-Hui, Wu Reng-Lai, Nan Wang, Quanhui Liu

Published Feb 1, 2010• 5 citations
Publisher

Abstract

The Schrödinger equation for a particle in the V-shaped potential decorated by a repulsive or attractive Dirac delta function interaction at the center is solved, demonstrating the crucial influence of point interaction on the even-parity states of the original system without decoration. As strength of the attraction increases, the ground state energy falls down without limit; and in limit of infinitely large attraction, the ground state approaches a singular state. Our analysis and conclusion can be readily generalized to any one-dimensional system a particle interacts with symmetrical potential plus the Dirac delta function interaction at the center.

DOI
10.1088/0253-6102/53/2/08

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The Schrödinger equation for a particle in the V-shaped potential decorated by a repulsive or attractive Dirac delta function interaction at the center is solved, demonstrating the crucial influence of point interaction on the even-parity states of the original system without decoration. As strength of the attraction increases, the ground state energy falls down without limit; and in limit of infinitely large attraction, the ground state approaches a singular state. Our analysis and conclusion can be readily generalized to any one-dimensional system a particle interacts with symmetrical potential plus the Dirac delta function interaction at the center.
Schr ? 为在一个排斥或漂亮的迪拉克三角洲功能相互作用在中心装饰的塑造 V 的潜力的一个粒子的 dinger 方程被解决，没有装饰，在原来的系统的平同等值的状态上表明点相互作用的关键影响。作为吸引力增加的力量，没有限制，扎根的州的精力掉下来；并且在无穷地大的吸引力的限制，扎根的状态走近一个单个州。我们的分析和结论能乐意地被概括到任何一个维的系统一个粒子在中心加迪拉克三角洲函数相互作用与对称的潜力交往。
Abstract In this paper, we propose a pedagogical presentation of the Lippmann–Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann–Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann–Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.
We obtain for the attractive Dirac δ-function potential in two-dimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in N−2 degrees of freedom (in the center of momentum frame) when there are N particles. The effective Hamiltonian for N−2 particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The three-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given.
The quantum mechanical problem of a particle moving in one dimension (1D) under the influence of a singular potential is reexamined. An attractive delta function potential of infinite strength and the 1D Coulomb potential is considered in particular. In both cases the singularity acts as an impenetrable barrier, thus dividing space into two independent regions. Analysis of the scattering states gives results in agreement with this conclusion. The implications that the bound states of the 1D Coulomb potential cannot be classified according to parity, even though the potential is symmetric, is also discussed herein.