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arXiv:1108.3135 · Progress in physics

The Heisenberg Uncertainty Principle and the Nyquist-Shannon Sampling Theorem

Pierre A. Millette

Published Aug 16, 2011• 10 citations
arXivHTMLPDF

Abstract

The derivation of the Heisenberg Uncertainty Principle (HUP) from the Uncertainty Theorem of Fourier Transform theory demonstrates that the HUP arises from the dependency of momentum on wave number that exists at the quantum level. It also establishes that the HUP is purely a relationship between the effective widths of Fourier transform pairs of variables (i.e. conjugate variables). We note that the HUP is not a quantum mechanical measurement principle per se. We introduce the Quantum Mechanical equivalent of the Nyquist-Shannon Sampling Theorem of Fourier Transform theory, and show that it is a better principle to describe the measurement limitations of Quantum Mechanics. We show that Brillouin zones in Solid State physics are a manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level. By comparison with other fields where Fourier Transform theory is used, we propose that we need to discern between measurement limitations and inherent limitations when interpreting the impact of the HUP on the nature of the quantum level. We further propose that while measurement limitations result in our perception of indeterminism at the quantum level, there is no evidence that there are any inherent limitations at the quantum level, based on the Nyquist-Shannon Sampling Theorem.

DOI
10.48550/arxiv.1108.3135

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The derivation of the Heisenberg Uncertainty Principle (HUP) from the Uncertainty Theorem of Fourier Transform theory demonstrates that the HUP arises from the dependency of momentum on wave number that exists at the quantum level. It also establishes that the HUP is purely a relationship between the effective widths of Fourier transform pairs of variables (i.e. conjugate variables). We note that the HUP is not a quantum mechanical measurement principle per se. We introduce the Quantum Mechanical equivalent of the Nyquist-Shannon Sampling Theorem of Fourier Transform theory, and show that it is a better principle to describe the measurement limitations of Quantum Mechanics. We show that Brillouin zones in Solid State physics are a manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level. By comparison with other fields where Fourier Transform theory is used, we propose that we need to discern between measurement limitations and inherent limitations when interpreting the impact of the HUP on the nature of the quantum level. We further propose that while measurement limitations result in our perception of indeterminism at the quantum level, there is no evidence that there are any inherent limitations at the quantum level, based on the Nyquist-Shannon Sampling Theorem.
The derivation of the Heisenberg Uncertainty Principle (HUP) from the Uncertainty Theorem of Fourier Transform theory demonstrates that the HUP arises from the dependency of momentum on a wave number that exists at the quantum level. It also establishes that the HUP is purely a relationship between the effective widths of Fourier transform pairs of variables (i.e. conjugate variables). We note that the HUP is not a quantum mechanical measurement principle per se. We introduce the Quantum Mechanical equivalent of the Nyquist-Shannon Sampling Theorem of Fourier Transform theory, and show that it is a better principle to describe the measurement limitations of Quantum Mechanics. We show that Brillouin zones in Solid State Physics are a manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level. By comparison with other fields where Fourier Transform theory is used, we propose that we need todiscern between measurement limitations and inherent limitations when interpreting the impact of the HUP on the nature of the quantum level. We further propose that while measurement limitations result in our perception of indeterminism at the quantum level, there is no evidence that there are any inherent limitations at the quantum level, based on the Nyquist-Shannon Sampling Theorem
Donoho and Stark have shown that a precise deterministic recovery of missing information contained in a time interval shorter than the time-frequency uncertainty limit is possible. We analyze this signal recovery mechanism from a physics point of view and show that the well-known Shannon-Nyquist sampling theorem, which is fundamental in signal processing, also uses essentially the same mechanism. The uncertainty relation in the context of information theory, which is based on Fourier analysis, provides a criterion to distinguish Shannon-Nyquist sampling from compressed sensing. A new signal recovery formula, which is analogous to Donoho-Stark formula, is given using the idea of Shannon-Nyquist sampling; in this formulation, the smearing of information below the uncertainty limit as well as the recovery of information with specified bandwidth take place. We also discuss the recovery of states from the domain below the uncertainty limit of coordinate and momentum in quantum mechanics and show that in principle the state recovery works by assuming ideal measurement procedures. The recovery of the lost information in the sub-uncertainty domain means that the loss of information in such a small domain is not fatal, which is in accord with our common understanding of the uncertainty principle, although its precise recovery is something we are not used to in quantum mechanics. The uncertainty principle provides a universal sampling criterion covering both the classical Shannon-Nyquist sampling theorem and the quantum mechanical measurement.
Heisenberg’s uncertainty principle (HUP) is a central concept in quantum mechanics that reveals inherent limitations in the measurement of microscopic particles and profoundly affects the understanding of the physical world. This paper reviews the physical significance of HUP and its applications and research progress in modern physics. The complexity of measurements in quantum systems is revealed by deriving the uncertainty relation and analysing the wavefunction collapse and the classicalisation of quantum states. This paper summarises the current experimental verification of the principle of measurement uncertainty in different quantum systems and discusses the research progress in improving measurement accuracy through quantum entanglement. Recent studies have shown that controlling quantum entangled states can effectively reduce measurement disturbances and thus improve the measurement accuracy of quantum systems without violating the principle of mismeasurement. Despite the many advances made, there are still challenges to further improve the measurement accuracy in complex quantum systems. Future research will continue to explore the potential for applications in this area.
Heisenberg’s uncertainty principle (HUP) is a central concept in quantum mechanics that reveals inherent limitations in the measurement of microscopic particles and profoundly affects the understanding of the physical world. This paper reviews the physical significance of HUP and its applications and research progress in modern physics. The complexity of measurements in quantum systems is revealed by deriving the uncertainty relation and analysing the wavefunction collapse and the classicalisation of quantum states. This paper summarises the current experimental verification of the principle of measurement uncertainty in different quantum systems and discusses the research progress in improving measurement accuracy through quantum entanglement. Recent studies have shown that controlling quantum entangled states can effectively reduce measurement disturbances and thus improve the measurement accuracy of quantum systems without violating the principle of mismeasurement. Despite the many advances made, there are still challenges to further improve the measurement accuracy in complex quantum systems. Future research will continue to explore the potential for applications in this area.
Heisenberg’s uncertainty principle (HUP) is a central concept in quantum mechanics that reveals inherent limitations in the measurement of microscopic particles and profoundly affects the understanding of the physical world. This paper reviews the physical significance of HUP and its applications and research progress in modern physics. The complexity of measurements in quantum systems is revealed by deriving the uncertainty relation and analysing the wavefunction collapse and the classicalisation of quantum states. This paper summarises the current experimental verification of the principle of measurement uncertainty in different quantum systems and discusses the research progress in improving measurement accuracy through quantum entanglement. Recent studies have shown that controlling quantum entangled states can effectively reduce measurement disturbances and thus improve the measurement accuracy of quantum systems without violating the principle of mismeasurement. Despite the many advances made, there are still challenges to further improve the measurement accuracy in complex quantum systems. Future research will continue to explore the potential for applications in this area.
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.