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量子力学、香农采样定理与数理统计学的数学联系

Zheng-Ling YANG

Published Dec 5, 2021• 0 citations
Publisher

Abstract

对于原子的基本结构，目前物理学认为电子围绕原子核运动，其不同轨道之间通常是分离的。对波函数、坍塌的解释有不同的观点。从信号处理的角度看：不妨假设电子围绕原子核做连续的复杂运动；利用奈奎斯特-香农采样定理（Nyquist-Shannon sampling theorem），可以把高速运动的微观粒子的连续行为的长期观察结果，采样成不连续的概率方式，即量子力学。“标准”的量子力学建立时（1926年前后），数学里的概率论和数理统计学还没有发展成熟。数学工具的不足，是引起波函数与坍塌解释不同看法的数学原因。

DOI
10.31219/osf.io/b7u62

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对于原子的基本结构，目前物理学认为电子围绕原子核运动，其不同轨道之间通常是分离的。对波函数、坍塌的解释有不同的观点。从信号处理的角度看：不妨假设电子围绕原子核做连续的复杂运动；利用奈奎斯特-香农采样定理（Nyquist-Shannon sampling theorem），可以把高速运动的微观粒子的连续行为的长期观察结果，采样成不连续的概率方式，即量子力学。“标准”的量子力学建立时（1926年前后），数学里的概率论和数理统计学还没有发展成熟。数学工具的不足，是引起波函数与坍塌解释不同看法的数学原因。
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
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.
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.