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Mathematical Physics and Computer Simulation

THE UNCERTAINTY PRINCIPLE FOR DIFFERENT SYSTEMS OF COORDINATE

АндрСй Π’Π°Π»Π΅Ρ€ΠΈΠ°Π½ΠΎΠ²ΠΈΡ‡ Павлов

Published Dec 27, 2024β€’ 0 citations
Publisher

Abstract

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(π‘₯ + 𝐴) equations for the same 𝑦 = 𝑓(π‘₯) graph for the only function, (the new equation of the same (𝑧, 𝑓(𝑝)) set of points for the 𝑝 = π‘₯, 𝑧 = 𝑦 complex variables). To prove the fact we can consider the 𝑦 = 𝑓(𝑝) equation for the new 𝑝 = π‘Ÿ + 𝐴 variable with π‘Ÿ = π‘₯; by definition of the 𝑝 and π‘Ÿ variables 𝑝 = π‘Ÿ + 𝐴, (the 𝑝 variable we consider in the primary coordinate system, the π‘Ÿ variable we consider in the new coordinate system with the (𝐴, 0) center of coordinate); for all the π‘Ÿ, π‘₯ variables π‘Ÿ = π‘₯ by definition of the radius-vectors in both coordinate systems, (π‘Ÿ is other designation of π‘₯). 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(π‘₯+𝑖𝑦) = 𝑓(βˆ’π‘₯+𝑖𝑦) by definition for the 𝑝 = π‘₯ + 𝑖𝑦 complex variable. In the new coordinate system the i constant is located on the OX axis instead of the π‘‚π‘Œ 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(𝑖π‘₯ + 𝑦) field in relation to the π‘₯ = 𝑦 diagonal is equal to the 𝑓(βˆ’π‘–(βˆ’π‘₯ + 𝑖𝑦)) field in relation to the 𝑖𝑂𝑋 axis.

DOI
10.15688/mpcm.jvolsu.2024.4.2

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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(π‘₯ + 𝐴) equations for the same 𝑦 = 𝑓(π‘₯) graph for the only function, (the new equation of the same (𝑧, 𝑓(𝑝)) set of points for the 𝑝 = π‘₯, 𝑧 = 𝑦 complex variables). To prove the fact we can consider the 𝑦 = 𝑓(𝑝) equation for the new 𝑝 = π‘Ÿ + 𝐴 variable with π‘Ÿ = π‘₯; by definition of the 𝑝 and π‘Ÿ variables 𝑝 = π‘Ÿ + 𝐴, (the 𝑝 variable we consider in the primary coordinate system, the π‘Ÿ variable we consider in the new coordinate system with the (𝐴, 0) center of coordinate); for all the π‘Ÿ, π‘₯ variables π‘Ÿ = π‘₯ by definition of the radius-vectors in both coordinate systems, (π‘Ÿ is other designation of π‘₯). 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(π‘₯+𝑖𝑦) = 𝑓(βˆ’π‘₯+𝑖𝑦) by definition for the 𝑝 = π‘₯ + 𝑖𝑦 complex variable. In the new coordinate system the i constant is located on the OX axis instead of the π‘‚π‘Œ 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(𝑖π‘₯ + 𝑦) field in relation to the π‘₯ = 𝑦 diagonal is equal to the 𝑓(βˆ’π‘–(βˆ’π‘₯ + 𝑖𝑦)) field in relation to the 𝑖𝑂𝑋 axis.
If we consider ( ) 2,kr tΞ¨ G to be the probability density for finding, at time t, particle 1 at particle k at and particle N at ,,1 … Gr …G ,kr ,Nr G a normalization condition analogous to Eq.(29.2) holds as (35.2) 1V* =ΨΨ∫ d The integral is defined in a space with 3N dimensions called the configuration space of the system, where a configuration point is given by 3N position coordinates 1 1 1, , , , , , , , , ,k k k N N N ,x y z x y z x y z… … and the volume element is NNNkkkNk dzdydxdzdydxdzdydxdddd ………… 1111 VVVV == Physical observables of the system will be represented by Hermitian operators, obtained from the classical representation in terms of the position and momentum variables as indicated by Postulate 2. Thus the many-particle Hamiltonian results from Eq.(2.32) as ( ) (βˆ‘ βˆ‘ = β‰  βŽ₯βŽ₯⎦ ⎀ ⎒⎒⎣ ⎑ ++βˆ‡βˆ’= k rrUtrU m H ,, 2 Λ† GGG= ) ) (35.3) where only acts on the coordinates of the kth particle, of mass The one-particle potential energy is due to external fields where this particle moves and kβˆ‡ .km( trU kk ,G( )jkkj rrU GG , gives the mutual interaction between the particles k and j. It is assumed that the position and momentum operators of different particles commute, and so do all the operators describing single-particle observables if they refer to different particles. The evolution of a many-particle system will be then formulated as a generalization of Eq.(30.6) in terms of given by Eqs.(35.1) and (35.3) HΛ†andΞ¨ Ξ¨=βˆ‚ Ξ¨βˆ‚ H t i Λ†= (35.4) If the Hamiltonian (35.3) does not depend explicitly on the time, it is usual to express the time dependence of the many-particle state function by means of the evolution operator defined as ( )tUΛ† ( ) ( ) ( )NkNk rrrtUtrrr G…G…GG…G…G ,,,,Λ†,,,,, 11 Ξ±Ξ¨=Ξ¨ (35.5) which is formally derived from Eq.(35.4) as ( ) ( ) ( )Λ†Λ† Λ†U ti U t i HU t t t Λ†Ξ± Ξ± Ξ± βˆ‚βˆ‚ ⎑ ⎀Ψ = Ξ¨ =⎣ βŽ¦βˆ‚ βˆ‚= = Ξ¨ It follows that (35.6) ( ) =/Λ†Λ† tHietU βˆ’= which is the analogue of Eq.(30.12). The evolution operator is unitary, Eq.(II.22), since ( ) ( )†ˆ Λ†U t U t IΛ†= (35.7) Substituting Eq.(35.5), one obtains from Eq.(35.4) that the stationary states of the Nparticle systems are given by the energy eigenvalue equation ( ) ( )1 1Λ† , , , , , , , ,k N k NH r r r E r r rΞ± Ξ± Ξ±Ξ¨ = Ξ¨G G G G G G… … … … (35.8) The solution to the N-particle problem is facilitated if we take into account the symmetries of many systems of physical interest, which allow us to define constants of motion and criteria for simplifying the form (35.3) of the Hamiltonian operator. An alternative to complex explicit calculations is to associate with specific symmetries time independent unitary operators which transform the state functions into so that Eq.(35.4) gives Λ† ,U Ξ¨ Λ† ,U Ξ¨ ( ) ( )†ˆ Λ† Λ† Λ† Λ† Λ† Λ†i U i U UH UHU Ut tβˆ‚ βˆ‚Ξ¨Ξ¨ = = Ξ¨ = Ξ¨βˆ‚ βˆ‚= = Λ† This is again a SchrΓΆdinger equation for the state UΛ† Ξ¨ with the new Hamiltonian †ˆ Λ† Λ† Λ†H UHUβ€² = If the Hamiltonian remains unchanged by the transformation, we have (35.9) †ˆ Λ† Λ† Λ† Λ† Λ†or , 0H UHU U H⎑ ⎀= ⎣ ⎦ = According to the evolution equation (29.43), this implies that so that U is a constant of motion. If the unitary transformation which leaves the Hamiltonian invariant, is a function of an observable, its constancy implies that the given observable is a constant in any state of the system.
Introduction Impenetrable bosons in one space dimension are a special case (at coupling constant c β†’ ∞) of the one-dimensional Bose gas (NS model) considered in detail in Chapter I. The results obtained there for the NS model can be specialized to the case of impenetrable bosons. It is to be said that all the formulΓ¦e are considerably simplified in this case. From the point of view of physics this is due to the fact that the Ξ΄-function potentials in the N -particle Hamiltonian (I.1.11) are now infinitely strong and the bosons cannot penetrate one another, so that the wave function should be equal to zero if the bosons' coordinates coincide. In this respect impenetrable bosons are rather similar to free fermions (see Appendix I.I). In this chapter correlation functions of impenetrable bosons are considered and their representations as Fredholm determinants of linear integral operators are given. The (very essential) simplification with respect to the general case is due to the fact that now all the auxiliary quantum fields can be set equal to zero (see IX.6), so that the kernels of these operators do not contain quantum fields. The determinant representation for the time-dependent correlator is easy to obtain in this case. Temperature correlation functions are considered as well as zero temperature ones.
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.
1. The analysis of single photon behaving in coordinate systems of various geometries has shown the following: Β· In Cartesian coordinate system the components of single photon eigen-vector are progressive and regressive plane waves. In other words, the photon behaving is characterized by coordinates and momentum, only. Β· In cylindrical coordinate system the components of single photon eigen-vector of are progressive and regressive plane waves, but only along z axis. In comparison to the results of analysis in Cartesian coordinates the fact that in x y planes photon oscillations are attenuated according to Β­1/2 law represents a generalization. This conclusion has been based on asymptotic behaving of Bessel functions. Β· The most authentic result is obtained by the analysis in spherical coordinates. Photon states do not depend on angles, and they are damped according to rΒ­1 law, where r is radius of the sphere. It has also turned out that orbital momentum of a free photon is equal to zero (this is understandable considering the fact that it moves along a straight line). The important element determining the behavior of a free photon is spin. One components of photon eigen-vector corresponds to spin Sz = 1/2 projection, while the other component corresponds to spin Sz = Β­1/2 projection. This was concluded on the basis of the fact that its eigen-vector components are Bessel functions having indices +1/2 and Β­1/2. The last result shows that linearization of photon Hamiltonian gives more complete picture of single photon than Kline-Gordon's approach. Concluding the exposed analysis we shall try to connect the results obtained in series of experimental investigation of photon orbital momentum (Beth, 1936; Leach, et al., 2002; Allen, et al., 1992; Allen, 1966; He, et al., 1995; Friese, et al., 1996; Markoski, et al., 2008; van Enk & Nienhuis, 2007; Santamato, et al., 1988; O'Neil, et al., 2002; Volke-Sepulveda, et al., 2002). We shall not describe all quoted experiments. Instead of it we shall describe the essential idea: the orbital momentum of photon was determines from the changes of