Quantum mechanics can answer the critical question in general signal processing: does any improvement beyond the Fourier resolution contradict the Heisenberg uncertainty principle ΔωΔt ≥ h? The correct answer requires a sharp distinction between the informational content and the resolving power. This chapter addresses the key question: why should quantum mechanics be used in processing time signals from NMR and MRS? It definitely should, because such time signals c(t) stem from purely quantum-mechanical phenomena. The function c(t) itself is the overall result of the induced transitions between two quantum-mechanically allowed states of a sample, perturbed by static, as well as varying magnetic fields and radio-waves tuned to a resonant frequency of the investigated spin-active nuclei with non-zero values of the magnetic moment.
Quantum mechanics can answer the critical question in general signal processing: does any improvement beyond the Fourier resolution contradict the Heisenberg uncertainty principle ΔωΔt ≥ h? The correct answer requires a sharp distinction between the informational content and the resolving power. This chapter addresses the key question: why should quantum mechanics be used in processing time signals from NMR and MRS? It definitely should, because such time signals c(t) stem from purely quantum-mechanical phenomena. The function c(t) itself is the overall result of the induced transitions between two quantum-mechanically allowed states of a sample, perturbed by static, as well as varying magnetic fields and radio-waves tuned to a resonant frequency of the investigated spin-active nuclei with non-zero values of the magnetic moment.
Abstract At the recent QSCP XIX, the author claimed a procedure of using a scaled Fourier transform (the scaling being determined by the detailed interaction and particle mass for a harmonic oscillator) to achieve simultaneous resolution of position and momentum greater than the standard Heisenberg value of 1/2. The procedure, while correct for digital signal processing, is, in fact, invalid for quantum mechanics. The purpose of this paper is simply to give the correct analysis of the uncertainty product, thereby clarifying the error made.
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.
Nuclear magnetic resonance (NMR) is a direct macroscopic manifestation of the quantum mechanics of the intrinsic angular momentum of atomic nuclei. It is best known for its extraordinary range of applications, which include molecular structure determination, medical imaging, and measurements of flow and diffusion rates. Most recently, liquid-state NMR spectroscopy has been found to provide a powerful experimental tool for the development and evaluation of the coherent control techniques needed for quantum information processing. This burgeoning new interdisciplinary field has the potential to achieve cryptographic, communications, and computational feats far beyond what is possible with known classical physics. Indeed, NMR has made the demonstration of many of these feats sufficiently simple to be carried out by high school summer interns working in our laboratory (see the last two authors). In this paper the basic principles of quantum information processing by NMR spectroscopy are described, along with several illustrative experiments suitable for incorporation into the undergraduate physics curriculum. These experiments are spin–spin interferometry, an implementation of the quantum Fourier transform, and the quantum simulation of a harmonic oscillator.
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
In radar analysis there exists an analogue of the Heisenberg uncertainty principle of quantum mechanics. Quantum mechanics stands here for the quantum-mechanical description, at a given instant of time, of a non-relativistic particle. These uncertainty principles suggest that there should exist a common mathematical structure behind both quantum mechanics and the theory of signals. It is the aim of the present article to establish that the concept of real Heisenberg nilpotent group Ã(ℝ) lies at the foundations of both of these fields. The crucial point is to endow the time-frequency plane ℝ ⊕ ℝ with the structure of a two dimensional real symplectic vector space so that it gets symplectomorphic to the tangent plane to the “Schrodinger coadjoint orbit” at the point 1 in the Kirillov orbit picture for the unitary dual of Ã(ℝ). Various applications of this geometric relationship between signal theory and nil potent harmonic analysis are pointed out. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.