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College Physics

Solution of one-dimensional motion of a particle in a constant force field

Yong Wang

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

Based on the equivalence between momentum representation and coordinate representation,an alternative method is presented,where the momentum representation and the Feynman's path integral are employed to calculate the propagator function of a particle in a constant force field,then the result is transformed into the coordinate representation. We present a way to solve the Feynman's path integral,which exhibits the high value of momentum representation if the solution of one- dimensional motion of a particle in a constant force field.

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Based on the equivalence between momentum representation and coordinate representation,an alternative method is presented,where the momentum representation and the Feynman's path integral are employed to calculate the propagator function of a particle in a constant force field,then the result is transformed into the coordinate representation. We present a way to solve the Feynman's path integral,which exhibits the high value of momentum representation if the solution of one- dimensional motion of a particle in a constant force field.
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.
The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum space, where the Schrödinger equation becomes an integral equation. To this end we discuss standard problems, namely, the free particle, the quantum motion under a constant potential, a particle interacting with a potential step, and the motion of a particle under a harmonic potential. What is not so standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. All the problems are solved within the momentum representation making no reference to the systems they correspond to in the coordinate representation.
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.