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arXiv:2502.02348

The Relationship Between the Number of Nodes in Wave Functions and Heisenberg's Uncertainty Principle

Tarek Yehia

Published Feb 4, 2025• 0 citations
arXivHTMLPDF

Abstract

This paper focuses on the complex relationship between Heisenberg's Uncertainty Principle and the nodal structure of wave functions in a variety of quantum systems including the quantum harmonic oscillator, the particle in a 1D box , and the particle on a ring. We argue that the uncertainty in conjugate variables, like location and momentum, is generally a function of the number of nodes. As our investigation reveals, the nature of this influence depends on the system. This paper demonstrates that Heisenberg's Uncertainty Principle is influenced by the nodal structure of wave functions and how the nature of this dependence is system-dependent.

DOI
10.48550/arxiv.2502.02348

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This paper focuses on the complex relationship between Heisenberg's Uncertainty Principle and the nodal structure of wave functions in a variety of quantum systems including the quantum harmonic oscillator, the particle in a 1D box , and the particle on a ring. We argue that the uncertainty in conjugate variables, like location and momentum, is generally a function of the number of nodes. As our investigation reveals, the nature of this influence depends on the system. This paper demonstrates that Heisenberg's Uncertainty Principle is influenced by the nodal structure of wave functions and how the nature of this dependence is system-dependent.
We study the influence of nodal structures in two-dimensional quantum mechanical densities on wave packet entanglement. This is motivated by our recent study [Entropy, 25, 970 (2023)], which showed that the mutual information derived from the momentum-space probability density of a coupled two-particle system exhibits an unusual time dependence, which is not encountered if the position-space density is employed in the calculation. In studying a model density, here, we identify cases where the mutual information increases with the number of nodes in the wave function and approaches a finite value, whereas in this limit, the linear correlation vanishes. The results of the analytical model are then applied to interpret the correlation measures for coupled electron-nuclear dynamics, which are treated by numerically solving the time-dependent Schrödinger equation.
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.
This article establishes the proof of the Schrdinger 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 Schrdinger 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 Schrdinger 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.
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.
Uncertainty relations and complementarity of canonically conjugate position and momentum observables in quantum theory are discussed with respect to some general coupling properties of a function and its Fourier transform. The question of joint localization of a particle on bounded position and momentum value sets and the relevance of this question to the interpretation of position-momentum uncertainty relations is surveyed. In particular, it is argued that the Heisenberg interpretation of the uncertainty relations can consistently be carried through in a natural extension of the usual Hilbert space frame of the quantum theory.