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UNITEXT for physics

The Heisenberg Uncertainty Principle and the Mathematics Behind

Salvatore Capozzıello, Wladimir-Georges Boskoff

Published Jan 1, 2021• 0 citations
Publisher
DOI
10.1007/978-3-030-86098-1_8

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Most physics books do not reveal clearly how the Heisenberg uncertainty principle was derived. This uncertainty comes from the consequence of the wave-particle duality of matter giving statement that position and momentum cannot be measured in the same time. This article tries to reveal mathematics background behind the expression the Heisenberg uncertainty using supported mathematics background such as Fourier transform, Fourier transform integral, the probability of Gaussian distribution and it ends up with the expression of wave function which describe the localized particle giving relation Heisenberg uncertainty principle.
Heisenberg’s uncertainty principle states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle (complementary variables) can be measured simultaneously. Heisenberg’s uncertainty principle has indubitable support, but the origin behind this principle is unexplained. If complementary variables of particles are considered as complex numbers—for example, in calculating particle position, a complex vector coordinate space is necessary instead of the Cartesian space—then the origin of lower limit of Heisenberg’s uncertainty principle emerges.
This is a multi-disciplinary paper. It borro ws ideas from mathemat ics, engineering software, and digital communicat ion engineering. Uncertainty principle is at the foundation of quantum mechanics. (A) It is well known that this principle is a consequence of Fourier transform (FT). The FT is based on infinity assumption. As infinity is not realistic and mean ingful in nature, and in engineering, we show that replacing in fin ity by any finite value changes the lower bound of the uncertainty principle to any desired accuracy number. (B) The paper points out, that uncertainty principle vio lates a very fundamental and well known concept in mathematics: the infinite d imensionality property of functions over finite intervals. (C) It is important to realize that no engineering experiment can prove any theory. Engineering is created out of objects of nature. Nature does not and cannot make any assumptions. Thus all engineering experimental setups will auto matically eliminate all assumptions from all theories. To establish this obvious and logical fact, we discuss many laws of nature, which modern microprocessor based engineering systems implement. Therefore it is not possible to prove uncertainty principle by any physical experiment, because the principle has many assumptions. (D) We exp lore several published proofs of uncertainty principle, including Heisenberg's and Operator theoretic, and analyze the assumptions behind them to show that this theory cannot be a law of nature. The paper ignores the relativistic effects.