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Canadian Journal of Physics

A new interpretation of the wave function: Reply

WAFIK A. WASSEF

Published Apr 1, 1990• 0 citations
Publisher

Abstract

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.

DOI
10.1139/p90-072

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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.
Abstract The Moyal equation describes the evolution of the Wigner function of a quantum system in the phase space. The right-hand side of the equation contains an infinite series with coefficients proportional to powers of the Planck constant. There is an interpretation of the Moyal equation as a quantum analogue of the classical Liouville equation. Indeed, if one uses the notion of the classical passage to the limit as the Planck constant tends to zero, then formally the right-hand side of the Moyal equation tends to zero. As a result, the Moyal equation becomes the classical Liouville equation for the distribution function. In this paper, we show that the right side of the Moyal equation does not explicitly depend on the Planck constant, and all terms of the series can make a significant contribution. The transition between the classical and quantum descriptions is related not to the Planck constant, but to the spatial scale. For a model quantum system with a potential in the form of a «quadratic funnel», an exact 3D solution of the Schrödinger equation is found and the corresponding Wigner function is constructed in the paper. Using trajectory analysis in the phase space, based on the representation of the right-hand side of the Moyal equation, it is shown that on the spatial microscale there is an infinite number of «trajectories» of the particle motion (thereby the concept of a trajectory is indefinite), and when passing to the macroscale, all «trajectories» concentrate around the classical trajectory.
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.
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.