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arXiv:1112.3674

Sharp and Infinite Boundaries in the Path Integral Formalism

Phillip Dluhy, Asim Gangopadhyaya

Published Dec 15, 2011• 3 citations
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Abstract

We revisit the analysis of sharp infinite potentials within the path integral formalism using the image method [1]. We show that the use of a complete set of energy eigenstates that satisfy the boundary conditions of an infinite wall precisely generates the propagator proposed in Ref. [1]. We then show the validity of the image method by using supersymmetric quantum mechanics to relate a potential without a sharp boundary to the infinite square well and derive its propagator with an infinite number of image charges. Finally, we show that the image method readily generates the propagator for the half-harmonic oscillator, a potential that has a sharp infinite boundary at the origin and a quadratic potential in the allowed region, and leads to the well known eigenvalues and eigenfunctions.

DOI
10.48550/arxiv.1112.3674

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We revisit the analysis of sharp infinite potentials within the path integral formalism using the image method [1]. We show that the use of a complete set of energy eigenstates that satisfy the boundary conditions of an infinite wall precisely generates the propagator proposed in Ref. [1]. We then show the validity of the image method by using supersymmetric quantum mechanics to relate a potential without a sharp boundary to the infinite square well and derive its propagator with an infinite number of image charges. Finally, we show that the image method readily generates the propagator for the half-harmonic oscillator, a potential that has a sharp infinite boundary at the origin and a quadratic potential in the allowed region, and leads to the well known eigenvalues and eigenfunctions.
The path integral formulation can reproduce the right energy spectrum of the harmonic oscillator potential, but it cannot resolve the Coulomb potential problem. This is because the path integral cannot properly take into account the boundary condition, which is due to the presence of the scattering states in the Coulomb potential system. On the other hand, the Sommerfeld quantization can reproduce the right energy spectrum of both harmonic oscillator and Coulomb potential cases since the boundary condition is effectively taken into account in this semiclassical treatment. The basic difference between the two schemes should be that no constraint is imposed on the wave function in the path integral while the Sommerfeld quantization rule is derived by requiring that the state vector should be a single-valued function. The limitation of the semiclassical method is also clarified in terms of the square well and δ(x)δ(x)δ(x) function potential models.
It is proposed that each of several analytical techniques used to evaluate the path integral when a spinless particle encounters an infinite potential barrier relies on more constraints than is necessary. We review the image, forbidden-paths, and eigenfunction-expansion methods, and the infinitesimal-propagator method is developed herein. It is then shown that the latter two methods contain assumptions leading to extra terms in the path-integral integrand which serve only to aid in the analytical evaluation of the path integral but which do not affect the outcome of that evaluation. The analytical arguments are further confirmed by a recently developed numerical method that, for a limited number of cases, is a direct evaluation of the path integral.
บทความนี้ศึกษาปัญหากำแพงศักย์อนันต์ในหนึ่งมิติโดยอาศัยวิธีอินทิกรัลตามเส้นทางของฟายน์แมน ได้คำนวณ ตัวแผ่กระจายสำหรับกำแพงศักย์อนันต์โดยวิธีอินทิกรัลตามเส้นทางแบบลากรางเกียน ผลเฉลยที่ได้อาศัยวิธีจุดภาพ กระจกเงาซึ่งสมมูลกับการรวมเส้นทางแบบฉบับ ตัวแผ่กระจายที่ได้อยู่ในรูปแบบที่กระชับ และจากตัวแผ่กระจายดังกล่าว สามารถนำไปคำนวณหาสถานะเจาะจงโมเมนตัมและระดับพลังงานที่เป็นไปได้ของอนุภาคที่อยู่ภายใต้อิทธิพลของกำแพง ศักย์อนันต์ได้ In this paper we study an infinite potential barrier problem in one dimension using Feynman path integral approach. The propagator for the infinite potential barrier is calculated by means of the Lagrangian path integral. This solution uses an image-point method equivalent to a sum over classical paths. The propagator is evaluated in a closed form, and from this propagator we obtained the momentum eigenstates and possible energy levels of a particle under the influence of an infinite potential barrier.