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Global Journal of Human Social Science

Exact Solution to the Schrodinger Equation with Manning-Rosen Potential Via Wkb Approximation Method

H. Louis

Published Apr 23, 2018• 0 citations
OpenAlex

Abstract

In quantum mechanics, exact solutions to equations play an important role because they contain a wealth of important information regarding the system under consideration. Here, we present the use of Wentzel-Kramers-Brillouin (WKB) approach toobtain the exact energy spectrum for Manning-Rosen potential and also, eigen energy solutions of special potential considered were also obtained.

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In quantum mechanics, exact solutions to equations play an important role because they contain a wealth of important information regarding the system under consideration. Here, we present the use of Wentzel-Kramers-Brillouin (WKB) approach toobtain the exact energy spectrum for Manning-Rosen potential and also, eigen energy solutions of special potential considered were also obtained.
The double-well potential is arguably one of the most important potentials in quantum mechanics, because the solution contains the notion of a state as a linear superposition of 'classical' states, a concept which has become very important in quantum information theory. It is therefore desirable to have solutions to simple double-well potentials that are accessible to the undergraduate student. We describe a method for obtaining the numerically exact eigenenergies and eigenstates for such a model, along with the energies obtained through the Wentzel–Kramers–Brillouin (WKB) approximation. The exact solution is accessible with elementary mathematics, though numerical solutions are required. We also find that the WKB approximation is remarkably accurate, not just for the ground state, but also for the excited states.
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.
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
The single harmonic oscillator and double-well potentials are important systems in quantum mechanics. The single harmonic oscillator is the paradigm in physics and is taught in nearly all beginner undergraduate classes, while the double-well potential illustrates the two important principles of quantum tunneling and linear superposition. While exact analytical solutions of the Schrödinger equation exist for both of these potentials, they are also employed to benchmark the use of approximate techniques, which may be the only recourse for more complicated potentials. In this paper, we review the Wentzel–Kramers–Brillouin (WKB) approximation for both these potentials. While this approximation is known for its accurate energies, we will instead emphasize how poor the WKB wave functions are. The inaccuracy of the WKB wave functions will then motivate us to adopt the lesser-known Modified Airy Function (MAF) approximation, which alleviates the deficiencies of the WKB wave functions. We will review the MAF solution to the simple harmonic oscillator potential and then apply the MAF to the double-well potential. We find accurate eigenvalues and, more importantly, very accurate wave functions. We conclude with the suggestion that an introduction to the MAF should be included in undergraduate courses to complement the WKB.
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.