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Jurnal Pendidikan Fisika dan Teknologi

Derivation and Physical Meaning of the Schrodinger Equation of a Particle in One Dimensional Space

Joko Krismanto Harianja

Published Jun 22, 2022• 0 citations
Publisher

Abstract

Research has been carried out to determine the physical meaning and derivation of the Schrodinger equation with the literature study research method. Based on the research results obtained through data processing in the form of information obtained from various literature sources, it can be concluded that the general solution of the Schrodinger equation is a wave function. In general, based on mathematical calculations by applying the technique of solving differential equations to the Schrodinger equation, it can be seen that this equation has many solutions, but not all of these solutions can be said to be functions that meet the wave function criteria. This wave function must be able to interpret the real physical criteria of a particle.

DOI
10.29303/jpft.v8i1.3455

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Research has been carried out to determine the physical meaning and derivation of the Schrodinger equation with the literature study research method. Based on the research results obtained through data processing in the form of information obtained from various literature sources, it can be concluded that the general solution of the Schrodinger equation is a wave function. In general, based on mathematical calculations by applying the technique of solving differential equations to the Schrodinger equation, it can be seen that this equation has many solutions, but not all of these solutions can be said to be functions that meet the wave function criteria. This wave function must be able to interpret the real physical criteria of a particle.
Time-independent and time-dependent forms of Schrodinger equations are presented in one and three dimensions along with their interpretations. Also, the time-independent version is shown to follow from the time-dependent version. For better conceptualization of wave motion, the general equation of wave motion is discussed and solved, explaining the physical significance of variables involved. Born interpretation of the wave function as a measure of the probability of finding the particle in a region of space at a given instant is described. The requirements to be fulfilled by the wave function for representing a physical world system are outlined.
The Schrödinger equation is a fundamental equation in physics for describing quantum mechanical behavior. It is a partial differential equation that shows how a wave function of a physical system evolves over time. In addition, it is associated with the kinetic energy and the potential energy, both of which contribute towards the total energy. This paper explains mathematical aspect of the Schrödinger equation. The concept of the WKB approximation has also been reviewed, which is an important method in finding the approximate solutions for the wave function. In addition, the paper also shows how to derive the transmission and reflection probabilities in relation to a one-dimensional quantum problem, called a “tunneling problem”, which occurs when a classical particle passes through a region where total energy is less than its potential energy. The WKB approximation can be used as a basis for formally writing down the exact solutions. Furthermore, it gives a high accuracy for the propagation of waves with high frequency or short wavelength.
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The Schrödinger equation serves as a pivotal differential equation, capturing the fundamental duality of particles as both matter and waves. Upon solving it, we derive a particle's wave function within a given potential, delineating the confines within which the particle operates. This wave function enables us to ascertain the "most probable" values for parameters such as position, momentum, and energy within the specified potential. In this study, we focus on the double-well potential in two dimensions, resembling a bowl-like structure. Specifically tailored equations, termed Equation 1 and Equation 2, stem from the general Schrödinger equation when applied to the prescribed conditions. This research aims to utilize these equations to describe a particle’s behavior within the ground and the first excited states of the double well potential, employing two distinct methodologies. The first methodology involves leveraging the variational method on Equations 1 and 2 to obtain approximate solutions, providing crucial inputs for our subsequent approach. The second methodology serves as entails an analytical solution of Equations 1 and 2. A comparative analysis of the outcomes from both methods is anticipated to ensure the coherence and validity of the variational method. This project integrates key principles and proficiencies from both physics and mathematics disciplines, offering a platform for honing practical skills in applying these methodologies within a real-world context.
Solving the Schrödinger equation may result in a wave function of a particle in a quantum system, which can afford the information with respect to the particle’s behavior. The Schrödinger equation is useful for examining probability density and determining a wave function of free particles. This research focuses on solving the Schrödinger equation using the Crank-Nicolson method in free particles. The Crank-Nicolson method is a method of solving partial differential equations in the form of the Schrodinger equation, this method is very stable and accurate in giving numerical results. The result indicates that probability density and the form of the wave function of free particles are identified by varying the v, N grid, and dt. When dt = 1, v = 1, and v = 2, and the N grid remains at a score of 100, we acquire the same forms of the wave function and probability density. And yet, when dt = 2, v =2, and the N grid remains at a score of 100, the form of the wave function and probability density is constricted in one area. The N grid and dt are the two most affecting factors on the three variants. Using the Crank-Nicolson method, we can determine the wave function and probability for free particles by varying the value of N grid, dt.