Andrey V. Pavlov The uncertainty principle for different systems of coordinate
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https://doi.org/10.15688/mpcm.jvolsu.2024.4.2
Andrey V. Pavlov
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics-1,
Moscow Institute of Radiotechnics,Electronics and Automatics — RTU
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,
https://orcid.org/0000-0002-1082-2222
Prosp. Vernadskogo, 78, MIREA, 117454 Moscow, Russian Federation
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(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 co-ordinates 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.
Key words: Periodicity, different coordinate systems, analytical functions, fields of the complex functions, moved functions, double representation of functions.
The uncertainty principle for different systems of coordinate by Andrey V. Pavlov is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 27 No. 4 2024, pp. 17-22
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