The time dependent Schrodinger equation is frequently 'derived' by postulating the energy E → i ℏ(∂/∂ t ) and momentum → (ℏ/ i ) operator relations. In the present 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: namely '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.
The time dependent Schrodinger equation is frequently 'derived' by postulating the energy E → i ℏ(∂/∂ t ) and momentum → (ℏ/ i ) operator relations. In the present 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: namely '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.
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.
It is shown how using the classical Hamilton-Jacobi equation one can arrive at the time-dependent wave equation. Although the former equation was originally used by E.Schroedinger to get the wave equation, we propose a different approach. In the first place, we do not use the principle of least action and, in addition, we arrive at the time-dependent equation, while Schroedinger (in his first seminal paper) used the least action principle and obtained the stationary wave equation. The proposed approach works for any classical Hamilton-Jacobi equation. In addition, by introducing information loss into the Hamilton-Jacobi equation we derive in an elementary fashion the wave equations (ranging from the Shroedinger to Klein-Gordon, to Dirac equations). We also apply this technique to a relativistic particle in the gravitational field and obtain the respective wave equation. All this supports 't Hooft's proposal about a possibility of arriving at quantum description from a classical continuum in the presence of information loss.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
Physics and the natural world have been studied by humans since the first person wondered ‘Why?’ The Schrödinger equation is important because it attempts to describe a relatively new (to us) and unfamiliar part of our universe, behavior of subatomic particles, and helps modern scientists uncover secrets that have proven useful in technology and explaining our origins. The time dependent Schrödinger equation can be derived from various starting points in Classical such as the equation of a wave, the time independent Schrödinger equation, and Hamilton-Jacobi equations. Why Quantum mechanics exists, useful applications, and cases when it fails or is inaccurate are often not understood or even easy to explain to people uninvolved in physical studies. Not only do I hope for someone to be more educated on the topic after encountering this project, but I, myself, have learned a great deal more about the advanced topics I am interested in. Class: Math 488 – Senior Capstone
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.