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Science, Technology, and Communication Journal.

Schrödinger's equation as a Hamiltonian system

Suhaivi Hamdan, Erwin Erwin, Saktioto Saktioto

Published Oct 30, 2022• 1 citation
Publisher

Abstract

This article describes the concept of the time-dependent Schrödinger equation (TDSE) as a review to add a detailed understanding of the steps to formulate TDSE which can be seen from the classical mechanic's concept as a mechanical wave function. In this article, a comprehensive approach to the concepts of momentum and energy in particles will be described using operators working on wave functions. The results show that the use of momentum and energy operators can show that TDSE is a Hamiltonian system.

DOI
10.59190/stc.v3i1.221

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This article describes the concept of the time-dependent Schrödinger equation (TDSE) as a review to add a detailed understanding of the steps to formulate TDSE which can be seen from the classical mechanic's concept as a mechanical wave function. In this article, a comprehensive approach to the concepts of momentum and energy in particles will be described using operators working on wave functions. The results show that the use of momentum and energy operators can show that TDSE is a Hamiltonian system.
The concept of time dependent Schrödinger equation (TDSE) illustrated in literature and even during class room teaching is mostly either complex or meant for advanced learners. This article is intended to enlighten the concept to the beginners in the field and further to improve knowledge about detailed steps for abstract mathematical formulation used which helps in understanding to derive TDSE using various tools and in more comprehensible manner. It is shown that TDSE may be derived using wave mechanics, time independent equation, classical & Hamilton-Jacobi’s equations. Similar attempts have been done earlier by some researchers. However, this article provides a comprehensive, lucid and well derived derivation, derived using various approaches, which would make this article unique.
This paper focuses on the principle of time dependent Schrodinger equation (TDSE) which simplified so many reviewed ideas, class room teaching, research ideologies and personal study meant for advanced knowledge. In this research work,  the enlightenment of the basic concept of Schrodinger wave equation to improve knowledge about simple ways for mathematical understanding in deriving TDSE using different technique in more comprehensive approach. The research shows clearly that TDSE can be derived using time independent equation, wave mechanics, classical & Hamilton-Jacobi’s equations. Different methods and ways by different researchers/scholars have been used in the past. In this paper, we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational. Surely, a particle is what a particle detector detects. This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation. However, the result provides a comprehensive well derived derivation, derived using various approaches, which would make this research a unique one from different areas of specializations.
The time-dependent Schrödinger equation (TDSE) is a wave equation that is used to understand the behavior of small particles such as electrons or nuclei. The equation must be "solved" in the sense that a function must be found that can be plugged into the equation and that leaves the left-hand side equal to the right-hand side. The TDSE can be understood by first studying the classical wave equation that is used to "solve" the behavior of a guitar string or a water wave. The TDSE is a bit different in that it equates a double space derivative with a single time derivative (the classical equation equates two double derivatives); it also implicitly includes the symbol i , which stands for the square root of -1. These differences give rise to different sorts of solutions, including complex solutions with real and imaginary parts. In the case of the TDSE, the solution or "wave" does not directly show the motion of the particle. Instead, it shows where the particle is likely to be found. More specifically, it is a probability wave for particle location. To obtain the precise probability for the particle to be found at a specific position x at a specific time t , the wave function Ψ ( x , t ) must be squared via a special procedure called taking the square modulus. The wave contains information about the frequency, wavelength, position, and momentum of the particle. In the following chapters, we will be showing how to find the wave solution in different circumstances (different potentials), and how to extract the information about momentum, etc., once the wave function is known.
The Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .
The Schrodinger and de Broglie hypothesis are directly derived from classical force and blank hypothesis. Thus, depending on the quantum equations the quantum of total energy relation (E = T + V ) and de Broglie relation (? = h=p). Accordingly, new time dependent Schrodinger equation is obtained. As a result, by using the wave function and nonrelativistic wave function beside plank hypothesis. Moreover, TDSE was obtained, thus by used additional kinetic energy and potential energy (classical equation). Finally, a result corresponding to TDSE in 3D was confirmed by using De Broglie hypothesis.