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Kuantum teknolojileri ve enformatik araştırmaları dergisi.

Derivative and partial integral methods and Gauss integral's indefinite integral solution and its use in wave function in quantum mechanics and exact solutions of the wave function depending on position and time

Dila Başpınar, Ela BAŞPINAR, Nesrin Başpınar, Umut Aldoğan

Published Oct 30, 2023• 0 citations
Publisher

Abstract

This study, , solution of Gaussian Integral with differential equation [18,19]; By using the partial integral method, the indefinite integral solution by taking x under the differential [29,30,31] and the indefinite integral solution by taking the Gaussian integral under the differential [33,38,39] both harmonic series and function solutions are found. Indefinite integral solution of Gaussian Integral [38] In Quantum Physics, the wave function solution f(x,α) in terms of α variable in x position and k space by substituting it in wave function [44,45] and in one-dimensional time dependent Schrödinger equation, wave function equation [66, 67] is found. When the wave function in direct space is differentiated by by the partial integral method without using the approximate value of w(k)[50] in space k, the general wave equation in k space by position [74] and approximately the wave function f(x,α) [80] is found. min in space k depending on location and time the exact solution of the wave equations [83,89] and the Taylor Series solution give the wave equation [87,90] in k space according to position and time.

DOI
10.70447/ktve.2239

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A new interpretation of the wave function: Reply

WAFIK A. WASSEF

I believe that the author of this note is confusing definite and indefinite integrals. Any indefinite integre 'I I can be reduced to the corresponding definite integral by the proper choice of the limits of integration and the constant of integration. The fact that [5] is a solution of [4] can be looked-up directly from any table of elementary integrals; see, for example, ref. 1. For finite values of z i t is easy to show that [3] and [4] are equivalent to the time-dependent Schrodinger's wave equation by taking the time derivative of both sides of [4]. Moreover, the limit z + 0 cannot be allowed, since from [3] this corresponds to an infinite energy E. The meaning of the unit of imaginary numbers in on the right-hand side of [4] is given immediately after this equation.

Canadian Journal of Physics•Apr 1, 1990•0 citations
This study, , solution of Gaussian Integral with differential equation [18,19]; By using the partial integral method, the indefinite integral solution by taking x under the differential [29,30,31] and the indefinite integral solution by taking the Gaussian integral under the differential [33,38,39] both harmonic series and function solutions are found. Indefinite integral solution of Gaussian Integral [38] In Quantum Physics, the wave function solution f(x,α) in terms of α variable in x position and k space by substituting it in wave function [44,45] and in one-dimensional time dependent Schrödinger equation, wave function equation [66, 67] is found. When the wave function in direct space is differentiated by by the partial integral method without using the approximate value of w(k)[50] in space k, the general wave equation in k space by position [74] and approximately the wave function f(x,α) [80] is found. min in space k depending on location and time the exact solution of the wave equations [83,89] and the Taylor Series solution give the wave equation [87,90] in k space according to position and time.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
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.
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.
A formulation of quantum mechanics is introduced based on a 2D2D2D-dimensional phase-space wave function p ⁣p(q,p)\text{\reflectbox{\text{p}}}\mkern-3mu\text{p}\left(q,p\right)pp(q,p) which might be computed from the position-space wave function ψ(q)ψ\left(q\right)ψ(q) with a transformation related to the Gabor transformation. The equation of motion for conservative systems can be written in the form of the Schrödinger equation with a 4D4D4D-dimensional Hamiltonian with classical terms on the diagonal and complex off-diagonal couplings. The Hamiltonian does not contain any differential operators and the quantization is achieved by replacing qqq and ppp with 2D2D2D-dimensional counterparts (q+q′)/2\left(q+q'\right)/2(q+q′)/2 and (p+p′)/2\left(p+p'\right)/2(p+p′)/2 and by using a complex-valued factor ei(q⋅p′−q′⋅p)/2e^{i\left(q\cdot p'-q'\cdot p\right)/2}ei(q⋅p′−q′⋅p)/2 in phase-space integrals. Despite the fact that the formulation increases the dimensionality, it might provide a way towards exact multi-dimensional computations as it may be evaluated directly with Monte-Carlo algorithms.
The Schrödinger equation for the wave functions is used to derive a linear partial differential equation for the time-dependent probability densities, ħ24m δ3P(x, t)δx3 = 2V(x) δPδx + V′P(x, t) − 2δε(x, t)δx − mδj(x, t)δt, where P(x, t), ε(x, t), j(x, t) are the time-dependent position, energy, and current densities, respectively. In the limiting case where the system is in the nth quantum state, it is shown that this partial differential equation reduces to the previously derived third-order ordinary differential equation for the probability position density.
I believe that the author of this note is confusing definite and indefinite integrals. Any indefinite integre 'I I can be reduced to the corresponding definite integral by the proper choice of the limits of integration and the constant of integration. The fact that [5] is a solution of [4] can be looked-up directly from any table of elementary integrals; see, for example, ref. 1. For finite values of z i t is easy to show that [3] and [4] are equivalent to the time-dependent Schrodinger's wave equation by taking the time derivative of both sides of [4]. Moreover, the limit z + 0 cannot be allowed, since from [3] this corresponds to an infinite energy E. The meaning of the unit of imaginary numbers in on the right-hand side of [4] is given immediately after this equation.