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Journal of Xinxiang University

Wave Function and Coordinate System

Yue Xiao-ping

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Discussing the relationship between wave function and coordinate system,this paper supposes that both single particle wave function and multiparticle wave function are related to coordinate system,which is always settled in the center of central field.With regard to multiparticle system wave function,it could be considered as the collection of different locations of a same particle,thus the constitution of its wave function could be perceived as the subjugation result of single wave function in examined location and the one in the other locations.The subjugation result of single wave function in unexamined locations could be considered as the normalization constant of single particle wave function in examined location.

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Discussing the relationship between wave function and coordinate system,this paper supposes that both single particle wave function and multiparticle wave function are related to coordinate system,which is always settled in the center of central field.With regard to multiparticle system wave function,it could be considered as the collection of different locations of a same particle,thus the constitution of its wave function could be perceived as the subjugation result of single wave function in examined location and the one in the other locations.The subjugation result of single wave function in unexamined locations could be considered as the normalization constant of single particle wave function in examined location.
According to the postulates of quantum mechanics, the state of a system is associated with a wave function that contains any measurable information on the system at any time. In this chapter we become familiar with wave functions and how they represent the position of particles within the system. Within the realm of quantum mechanics, the position of particles is not deterministic. It is defined by a probability distribution. The wave function is a position-dependent complex-valued amplitude, whose absolute value squared is identified with the probability density for locating the particle in the position space. This identification of the wave function with a probability amplitude imposes some limitation. Particularly, for a closed system in which the particles are bound, the wave function must be proper (square integrable) and normalizable. These properties are discussed and demonstrated for different coordinate systems.
Abstract The main provisions of the construction of the single-particle wave function of a photon in the coordinate representation are expounded and its application to the description of single-photon interference phenomena is discussed. The connection between the concepts of electromagnetic field strengths used in quantum electrodynamics and the characteristics of wave packets in photon quantum mechanics is substantiated. For illustration, we consider the relationship between the structure of the photon wave function in coordinate representation and the polarization of photon in the formalism of quantum transition amplitudes. The meaning of the concept of linear polarization of a photon in the state of a wave packet is explained. The wave function is constructed in the coordinate representation of the state of two entangled photons at the exit from a nonlinear crystal by modeling the polarization vectors and Gaussian momentum distribution for each of the photons.
The conditions for observation of the particle coordinates, required by logic of the Special Relativity and filtering the quantum field effects, are described. A general relation between the corresponding density of probability and the wave function is found. It is a relativistic invariant describing probability of the particle emergences in space -time. This density is concretized for bosons, both scalar and vector (including photon), charged and neutral, also electron. The Heisenberg's uncertainty relations have been approved in regards to relativistic particle. As applied to the quantum field, this new construction is transformed to new characteristic of the particles distribution in space-time, which complete distribution throughout impulses. The operators of these distributions and the invariant relativistic description for free quantum fields have been obtained. These new properties of the particles and fields are proposed for experimental investigations.
The hydrogen atom can be considered a two-particle system, an electron and a proton, with the Coulomb potential acting between them. This chapter shows how to solve the Schrodinger equation for the wave function of the two-particle system. It introduces the coordinates of the center of mass, which will allow us to split the Schrodinger equation into two independent equations, one for the center of mass motion and another for the relative motion of the electron and proton. The left- hand side of the Schrodinger equation depends solely on the spatial variables, whereas the right-hand side depends solely on time. The wave function can be written as a product of the spatial and time-dependent parts. It is very similar in form to the Schrodinger equation for the hydrogen atom except it involves two Laplacians. Since the Laplacians act on separate variables, the chapter helps to solve the Schrodinger equation using the method of separate variables.
The relationship between classical and quantum mechanics is explored in an intuitive manner by the exercise of constructing a wave in association with a classical particle. Using special relativity, the time coordinate in the frame of reference of a moving particle is expressed in terms of the coordinates in the laboratory frame of reference in order to provide an initial spatiotemporal function to work from in initiating the development of a quantum wave. When temporal periodicity is ascribed to the particle, a provisional spatiotemporal function for a particle travelling at constant velocity manifests itself as an running wave characterized by parameters associated with the moving particle. A wave description for bidirectional motion is generated based on an average time coordinate for a combination of oppositely directed elementary running waves, and the resulting spatiotemporal function exhibits wave behavior characteristic of a standing wave. Ascribing directional orientation to the intrinsic periodicity of the particle introduces directional sub-states; variations in the relative number of sub-states as a function of angle in combined states lead to spatially varying magnitudes for the associated waves. Further analysis leads to full mathematical expression for all waves representing free particle motion. A generalization for particles subject to force fields enables us to develop a governing differential equation identical in form to the Schroedinger equation.
We consider the analytical 𝑧 = 𝑓(𝑝) functions in different coordinate systems. The centers of coordinates of the new systems are located in the (𝐴, 0) points, with the r variable. With point of view of the new coordinate system we obtain some new 𝑦 = 𝑓(𝑥 + 𝐴) equations for the same 𝑦 = 𝑓(𝑥) graph for the only function, (the new equation of the same (𝑧, 𝑓(𝑝)) set of points for the 𝑝 = 𝑥, 𝑧 = 𝑦 complex variables). To prove the fact we can consider the 𝑦 = 𝑓(𝑝) equation for the new 𝑝 = 𝑟 + 𝐴 variable with 𝑟 = 𝑥; by definition of the 𝑝 and 𝑟 variables 𝑝 = 𝑟 + 𝐴, (the 𝑝 variable we consider in the primary coordinate system, the 𝑟 variable we consider in the new coordinate system with the (𝐴, 0) center of coordinate); for all the 𝑟, 𝑥 variables 𝑟 = 𝑥 by definition of the radius-vectors in both coordinate systems, (𝑟 is other designation of 𝑥). The consideration of the equations results in periodicity of the 𝑓(𝑝) function. The same result we obtain for the complex 𝐹(𝑝) field, where 𝐹(𝑥+𝑖𝑦) = 𝑓(−𝑥+𝑖𝑦) by definition for the 𝑝 = 𝑥 + 𝑖𝑦 complex variable. In the new coordinate system the i constant is located on the OX axis instead of the 𝑂𝑌 axis. In the system of coordinates the new equation of the (𝑧, 𝑓(𝑝)) set of points (graph) is the same as in the initial system of coordinates. It is proved, that the 𝑓(𝑖𝑥 + 𝑦) field in relation to the 𝑥 = 𝑦 diagonal is equal to the 𝑓(−𝑖(−𝑥 + 𝑖𝑦)) field in relation to the 𝑖𝑂𝑋 axis.