This paper presents a mathematical approach, based on the spectral analysis and Fourier transformation to derive the principle of uncertainty. The approach presented in this paper is found to be different from those appeared so far in the literature. It shows that the uncertainty principle is, in fact, an aliasing phenomenon and emerges merely from the quantization effect. The concept of ‘certainty’ is also introduced and discussed. It is shown that observation rates higher than the speed of a particle are the only possible approaches to the absence of uncertainty in the measurements of the observable quantities ofthe entity. This would imply that for particles travelling at the velocity of light, data acquisition rates higher than the speed of light should be required. Therefore, the uncertainty principle is related to the theory of relativity and exists so far as the velocity of light poses as an upperlimit universal constant. Otherwise, it would render to ‘certainty’ if our observation abilities could escalade at rates above the speed of light. This is to say that the uncertainty can be an inherent intrinsic property grounded in the universal speed limitation, which also subdues our observation abilities too; else demanding challenges on Einstein’s definition of simultaneity, and hence its consequences, has to be expected.
This paper presents a mathematical approach, based on the spectral analysis and Fourier transformation to derive the principle of uncertainty. The approach presented in this paper is found to be different from those appeared so far in the literature. It shows that the uncertainty principle is, in fact, an aliasing phenomenon and emerges merely from the quantization effect. The concept of ‘certainty’ is also introduced and discussed. It is shown that observation rates higher than the speed of a particle are the only possible approaches to the absence of uncertainty in the measurements of the observable quantities ofthe entity. This would imply that for particles travelling at the velocity of light, data acquisition rates higher than the speed of light should be required. Therefore, the uncertainty principle is related to the theory of relativity and exists so far as the velocity of light poses as an upperlimit universal constant. Otherwise, it would render to ‘certainty’ if our observation abilities could escalade at rates above the speed of light. This is to say that the uncertainty can be an inherent intrinsic property grounded in the universal speed limitation, which also subdues our observation abilities too; else demanding challenges on Einstein’s definition of simultaneity, and hence its consequences, has to be expected.
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.
One of the most fascinating principles in quantum mechanics must be Heisenberg’s uncertainty principle, which can be briefly stated as follows: every physical observation cannot be precisely determined without some degree of error or uncertainty. And it is by no means can one use the principle within the limit of certainty region, as will be shown in this Letter. Two of the most important pillars in modern physics must be Einstein’s relativity theory and Schr dinger’s contribution to quantum mechanics. Yet, there is a profound connection between these discoveries by means of the uncertainty relationship, in which we shown that the observation of a high-speed object is conceivable if the speed of the observer keeps up with object’s speed.
In this paper, using two different formulations, we attempted to first provide an overview of mathematical interpretations of Heisenberg's uncertainty principle. This was done in order to emphasise the mathematical justification for the uncertainty principle's existence. First, we went over in detail the matrix formulation found in Schrodinger's original paper on uncertainty. Next, we reviewed the mathematical proof of the uncertainty principle in a Fourier series formulation and examined the results. We attempted to provide a comprehensive overview of the modified forms of the uncertainty relation in the Quantum Gravitational, Relativistic, and Entropic domains along with other variants after realizing that the most well-known uncertainty relation is not universal. All of these concepts also demonstrate a great deal of potential for use in the emerging new technologies. Our goal was to gather all of these dispersed variations of the uncertainty principle in one location. The current state of these theories, as well as their applications and limitations, are thoroughly discussed.
The Heisenberg equation may not be as well-known as Einstein’s e=mc^2. But the Heisenberg equation’s implications that there is an irreducible uncertainty with regard to the position of an object are no less intriguing than the implications of Einstein’s relativity equation that time slows to a stop when the relative velocity of an object approaches the speed of light. Using elementary mathematics and science, this paper begins by examining the physics of motion of an everyday object such as a tennis ball. The paper shows that the same physics that applies to everyday objects in fact also applies when it comes to subatomic particles, such as an electron in motion. And by taking gradual incremental steps in developing the mathematics of the underlying physics, the paper leads to the conclusion that the universe is inherently uncertain, that it is impossible to know the exact position of any object.
The Heisenberg equation may not be as well-known as Einstein’s e=mc^2. But the Heisenberg equation’s implications that there is an irreducible uncertainty with regard to the position of an object are no less intriguing than the implications of Einstein’s relativity equation that time slows to a stop when the relative velocity of an object approaches the speed of light. Using elementary mathematics and science, this paper begins by examining the physics of motion of an everyday object such as a tennis ball. The paper shows that the same physics that applies to everyday objects in fact also applies when it comes to subatomic particles, such as an electron in motion. And by taking gradual incremental steps in developing the mathematics of the underlying physics, the paper leads to the conclusion that the universe is inherently uncertain, that it is impossible to know the exact position of any object.
The principle of uncertainty is one of the essential principles of quantum mechanics as it outlines that there is a reflective limit to the measurement of microscopic particle motion. However it is impossible to measure at once and accurately both the position and momentum data of any microscopic particle no matter what means are used to do so. This paper starts by discussing the notion and mathematical derivation of the uncertainty principle as put forth by Heisenberg explaining the physical relevance of the principle. It introduces the natural uncertainty the principle introduces to the dynamics of particles in the microscopic world, and the significant part it plays in the theory of quantum mechanics. The article overviews the explanations of the principle on phenomena including atomic stability and scanning tunneling microscopes, and the interference of measurement actions on the states of particles and the shortcomings of measurement precision. According to the wave-particle duality of quantum mechanics, this article gives a required theoretical basis of microscopic phenomena. Likewise, the use of the research theory on the uncertainty principle by distinguished scholars in the physics fraternity helps the readers in achieving the knowledge in related subjects more easily.