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.
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.
A mathematical uncertainty principle is an inequality or uniqueness theorem concerning the joint localization of a function or system and its spectrum. A Fourier uncertainty principle is thus a statement that a function and its Fourier transform cannot both decay too rapidly. The most familiar form is the Wiener-Heisenberg inequality, 5.1 $$ \left\| {(x - x_0 )f(x)} \right\|_2 \left\| {(\xi - \xi _0 )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (\xi )} \right\|_2 \geqslant \frac{{\mathop {\left\| f \right\|}\nolimits_2^2 }} {{4\pi }},$$ which becomes an equality only for multiples of Gaussians of the form exp(−πα(x − x0)2 + 2πiξ0x). In this chapter, D will denote the normalized differentiation or momentum operator D = (1/2πi)d/dx. The inequality (5.1) is often written 5.2 σX(f)σD(f)≡∥Xf−⟨Xf,f⟩f∥2∥Df−⟨Df,f⟩f∥2⩾1, where X is the multiplication operator f(x) ↦ xf(x). Barnes [22] attributes the first proof of (5.1) to a lecture delivered by Wiener in Hilbert’s mathematical physics seminar in Göttingen in 1925. In fact, Wiener discusses this lecture, though not the explicit proof of (5.1), at length in his autobiography (see [365], p. 105). What distinguished Wiener’s approach, however, was his macroscopic interpretation of (5.1), a point of view that led to further important achievements, including the Landau, Pollack and Slepian “Bell Labs” uncertainty principles in Chapter 3
This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
Certain results that have been predicted by Quantum Mechanics (QM) theory are not always supported by experiments. This defines a deep crisis in contemporary physics and, in particular, quantum mechanics. We believe that, in fact, the mathematical apparatus employed within today's physics is a possible reason. In particular, we consider the concept of infinity that exists in today's mathematics as the root cause of this problem. We have created Observer's Mathematics that offers an alternative to contemporary mathematics. This paper is an attempt to relay how Observer's Mathematics may explain some of the contradictions in QM theory results. We consider the Hamiltonian Mechanics, Newton equation, Schrodinger equation, two slit interference, wave-particle duality for single photons, uncertainty principle, Dirac equations for free electron in a setting of arithmetic, algebra, and topology provided by Observer's Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to solution of these problems are provided.
Applications of interval computations usually assume that while we only know an interval containing the actual (unknown) value of a physical quantity, there is the exact value of this quantity, and that in principle, we can get more and more accurate estimates of this value. Physicists know, however, that, due to uncertainty principle, there are limitations on how accurately we can measure the values of physical quantities. One of the important principles of modern physics is operationalism -- that a physical theory should only use observable properties. This principle is behind most successes of the 20th century physics, starting with relativity theory (vs. un-observable aether) and quantum mechanics. From this viewpoint, it is desirable to avoid using un-measurable exact values and to modify the mathematical formalisms behind physical theories so that they explicitly only take objective uncertainty into account. In this paper, we describe how this can be done for objective interval uncertainty.
Previously, Bennet and Feynman asked if Heisenberg's uncertainty principle puts a limitation on a quantum computer (Quantum Mechanical Computers, Richard P. Feynman, Foundations of Physics, Vol. 16, No. 6, p597-531, 1986). Feynman's answer was negative. In this paper, we will revisit the same question for the discrete time Fourier transform uncertainty principle. We will show that the discrete time Fourier transform uncertainty principle plays a fundamental role in showing that Shor's type of quantum algorithms has efficient running time and conclude that the discrete time uncertainty principle is an aid in our current formulation and understanding of Shor's type of quantum algorithms. It turns out that for these algorithms, the probability of measuring an element in some set T (at the end of the algorithm) can be written in terms of the time-limiting and band-limiting operators from finite Fourier analysis. Associated with these operators is the finite Fourier transform uncertainty principle. The uncertainty principle provides a lower bound for the above probability. We will derive lower bounds for these types of probabilities in general. We will call these lower bounds quantum algorithm uncertainty principles or QAUP. QAUP are important because they give us some sense of the probability of measuring something desirable. We will use these lower bounds to derive Shor's factoring and discrete log algorithms.
This article establishes the proof of the Schrödinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrödinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrödinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.