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.
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.
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.
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.
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.
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 use of a total time derivative of operators, that depends on the time evolution of the wave function as well as on any intrinsic time dependence in the operators, simplifies the formal development of quantum mechanics and allows its development to more closely follow the corresponding development of classical mechanics. We illustrate the use of the total time derivative for a free particle, the linear potential, the harmonic oscillator, and the repulsive inverse square potential. In these cases, operators whose total time derivative is zero can be found and yield general properties of wave packets and several useful time-dependent solutions of Schrödinger’s equation, including the propagator.
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 .