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Advances in physics theories and applications

Time Dependant Momentum Operator and Time Evolution of Free Quantum Particles

Mustafa Erol, Atakan Çoban

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

In spite of outstanding achievement of the quantum theory, the theory is under never-ending judgment due to probabilistic character and certain unresolved paradoxes arising from the very fundamentals. Additionally, it is debated that a more sophisticated theory beyond the quantum theory emerges as a need for a better understanding of the atomic and sub atomic worlds. This paper, for that reason, focuses on the very fundamental structure of the quantum theory such as the physical and philosophical meanings of the wave functions, the probability and the Schrödinger Wave Equation (SWE). Specifically deals with the time evolution of the free quantum particles. The work just arises from the difference of the classical and quantum waves, specifically the comparison of the time and space dependence of classical waves and Schrödinger waves, for the free space, indicates that time dependence shows a clear dissimilarity. In order to tackle the problem, a time dependent momentum operator, for the first time, is defined and consequently an alternative TDSWE is straightforwardly derived and its philosophical implications have been discussed. Additionally, time and position dependence of the probability functions, relating free particles, have been physically and philosophically discussed and some alternative approaches and comments are forwarded. Finally, the approaches, comments and mathematical equations are in good agreement with the standard quantum theory and interestingly go further possibly tracing some signs of new pathways to resolve the mysterious quantum world. Keywords: Schrödinger wave equation, fundamentals of quantum mechanics, momentum operator, quantum probabilities, time evolution of quantum systems.

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This study focuses on the temporal evolution of quantum systems based on epistemological and ontological arguments and reconsiders some fundamental concepts of quantum theory. Initially, the time dependent Schrödinger wave equation (TDSWE) is criticized for the lack of a potential energy term and thereafter, a novel time dependent momentum operator is defined which eventually leads to an alternative TDSWE, including a potential energy term that remains in harmony with current theory. The wave nature is then generally examined by comparing time evolutions of classical and Schrödinger waves, dissimilarities then carefully inspected and ultimately a novel second order TDSWE is suggested. Finally, time evolution of the probability density is discussed in light of this new approach, and additionally some further insights and conclusions are briefly outlined that conform with the standard theory.

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Introduction

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We consider the two basic theories of the standard physics: (1) the theory of relativity, as a four-dimensional time-space description of our universe, with the invariance of the time-space interval which leads to the Lorentz transformation, and (2) quantum mechanics, based on the Schrödinger equation, with a solution called wave function. However, we find that this equation is very unsatisfactory, describing fully unlocalized free particles, any initially localized particle rapidly spreading in space, in disagreement with the fundamental Hamilton equations. In agreement with the Hamilton equations, we describe a quantum particle by propagation wave functions in the coordinate and momentum spaces, with the Lagrangian in the time dependent phase instead of the Hamiltonian coming from the Schrödinger equation. With the relativistic Lagrangian, we obtain wave functions describing invariant distributions propagating in space.

Open quantum physics and environmental heat conversion into usable energy•Feb 7, 2022•0 citations
In spite of outstanding achievement of the quantum theory, the theory is under never-ending judgment due to probabilistic character and certain unresolved paradoxes arising from the very fundamentals. Additionally, it is debated that a more sophisticated theory beyond the quantum theory emerges as a need for a better understanding of the atomic and sub atomic worlds. This paper, for that reason, focuses on the very fundamental structure of the quantum theory such as the physical and philosophical meanings of the wave functions, the probability and the Schrödinger Wave Equation (SWE). Specifically deals with the time evolution of the free quantum particles. The work just arises from the difference of the classical and quantum waves, specifically the comparison of the time and space dependence of classical waves and Schrödinger waves, for the free space, indicates that time dependence shows a clear dissimilarity. In order to tackle the problem, a time dependent momentum operator, for the first time, is defined and consequently an alternative TDSWE is straightforwardly derived and its philosophical implications have been discussed. Additionally, time and position dependence of the probability functions, relating free particles, have been physically and philosophically discussed and some alternative approaches and comments are forwarded. Finally, the approaches, comments and mathematical equations are in good agreement with the standard quantum theory and interestingly go further possibly tracing some signs of new pathways to resolve the mysterious quantum world. Keywords: Schrödinger wave equation, fundamentals of quantum mechanics, momentum operator, quantum probabilities, time evolution of quantum systems.
This study focuses on the temporal evolution of quantum systems based on epistemological and ontological arguments and reconsiders some fundamental concepts of quantum theory. Initially, the time dependent Schrödinger wave equation (TDSWE) is criticized for the lack of a potential energy term and thereafter, a novel time dependent momentum operator is defined which eventually leads to an alternative TDSWE, including a potential energy term that remains in harmony with current theory. The wave nature is then generally examined by comparing time evolutions of classical and Schrödinger waves, dissimilarities then carefully inspected and ultimately a novel second order TDSWE is suggested. Finally, time evolution of the probability density is discussed in light of this new approach, and additionally some further insights and conclusions are briefly outlined that conform with the standard theory.
Time evaluation of wave functions for any quantum mechanical system/particle is essential nevertheless quantum mechanical counterpart of the time dependant classical wave equation does simply not appear. Epistemologically and ontologically considered time dependant momentum operator is initially defined and an Alternative Time Dependant Schrodinger Wave Equation (ATDSWE) is plainly derived. Consequent equation is primarily solved for the free particles, in a closed system, signifying a good agreement with the outcomes of the ordinary TDSWE. Free particle solution interestingly goes further possibly tracing some signs of new pathways to resolve the mysterious quantum world.
Present study focuses on some foundational problems of quantum theory specifically deals with the concept of probability density and relating introductory problems. In this sense, the work initially investigates the origins of the general probability theory and re-examines the concepts of spatial and temporal probability densities based on genuine epistemological and ontological arguments. In order to tackle the foundational problems, standard theory is primarily memorised and criticized scientifically and philosophically in terms of foundationally disappearing term of time dependent potential energy within the time and space dependent Schrödinger wave equation. Based on those arguments, the problematic inconsistency between the spatial and temporal probability density functions is underlined. Given the problem, an original approach previously suggested, is concisely described and extended to resolve the existing problem. The novel approach, based on a novel time dependent Schrödinger wave equation, resolves the discrepancy with the classical wave equation and also leads to time dependent temporal probability densities even for the time free potential energies. Novel temporal probability density function is also normalized and has a fluctuation period of around 10-16 s which is very short compared to the atomic time scales.
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.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
We consider the two basic theories of the standard physics: (1) the theory of relativity, as a four-dimensional time-space description of our universe, with the invariance of the time-space interval which leads to the Lorentz transformation, and (2) quantum mechanics, based on the Schrödinger equation, with a solution called wave function. However, we find that this equation is very unsatisfactory, describing fully unlocalized free particles, any initially localized particle rapidly spreading in space, in disagreement with the fundamental Hamilton equations. In agreement with the Hamilton equations, we describe a quantum particle by propagation wave functions in the coordinate and momentum spaces, with the Lagrangian in the time dependent phase instead of the Hamiltonian coming from the Schrödinger equation. With the relativistic Lagrangian, we obtain wave functions describing invariant distributions propagating in space.