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Journal of Southwest Jiaotong University

WKB METHOD FOR SOME NONLINEAR EQUATIONS: A SPECIAL CASE

Pedro Pablo Cárdenas Alzate, José Rodrigo González Granada, Fernando Mesa, Pedro Pablo Cárdenas Alzate, José Rodrigo González Granada, Fernando Mesa

Published Jan 1, 2023• 0 citations
Publisher

Abstract

This paper introduces a novel application of the Wentzel-Kramers-Brillouin (WKB) method to the Schrödinger equation, with the aim of incorporating certain nonlinear modifications that consider gravitational interaction within the wave function. By extending the traditional Schrödinger equation to account for gravitational effects, a new framework emerges, leading to two different sets of nonlinear differential equations for the gravitational potential, denoted as U . It is important to emphasize that these equations exhibit different characteristics and behaviors depending on the sign of U . The main objective of this study is to present a WKB method for the Schrödinger equation, which involves gravitational interactions in the wave function.The authors apply this method to derive two separate nonlinear differential equations for the gravitational potential U , considering both positive and negative values of U. The results reveal the intricate relationship between gravitational interactions and the behavior of the wave function, providing valuable insights into the dynamics of quantum systems under the influence of gravity. The presented WKB method extends our understanding of how gravitational effects affect quantum phenomena. As a novelty in this work, the integration of the WKB method with the Schrödinger equation is shown to address gravitational interactions. This unique approach opens new avenues for exploring quantum systems in the presence of gravity, which could lead to a deeper understanding of the fundamentals of quantum physics. Keywords: Nonlinear Differential Equation, Schrödinger Equation, Successive Approximations, The Wentzel-Kramers-Brillouin Method DOI： https://doi.org/10.35741/issn.0258-2724.58.4.66

DOI
10.35741/issn.0258-2724.58.4.66

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In this chapter we study another very important and powerful approximation method, the so-called eikonal approximation or Wentzel-Kramer-Brillouin (WKB) method, which is used in studying quantum mechanical systems that are subjected to complicated potentials. This method can be used to find approximate solutions to differential equations where the highest derivative is multiplied by a small parameter. The WKB method gives approximate solution to the Schrödinger equation irrespective of its complicated potential and is mostly used to study one-dimensional systems and also the radial equations for higher-dimensional systems with rotational symmetry.
Abstract This paper introduces a widely used approximation method, the Wentzel-Kramers-Brillouin (WKB) method, for solving the Schrodinger equation, sets up a theoretical framework to derive the general solutions of the method, and carries out the application of the approximation method in specific cases to verify the validity and practicality of the method. Throughout the study, numerical methods, such as the Simpsons method, and programming tools, such as Python, are used to process sophisticated calculations. The final approximated solutions for the Schrodinger equation are able to demonstrate the state of a particle in an isolated quantum system. All the presented studies are based upon the knowledge and skillsets of a high school student.
We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation for solving the Schrödinger equation. Based on an ansatz for the wave function, subject to an auxiliary condition on its first derivative, we obtain a set of coupled differential equations that decouple, via an alternating perturbation method, into the well-known Bremmer series. The perturbation improves the wave function amplitudes alternately, and its phase is refined through recursive diagonalization. From this construction, we derive a general quantization formula that encodes the geometric-optics-like physics of the system. When the ratio of the differential reflection coefficient to the classical momentum remains constant, the formula reduces to a closed-form quantization condition consistent with that obtained by re-summing the WKB series to all orders.
The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The Schrödinger equation is solved to find the penetration of a square potential barrier. The WKB method is applied to obtain a transmission coefficient for penetration through a barrier.