Concepts in Magnetic Resonance Part A

NMR and the uncertainty principle: How to and how not to interpret homogeneous line broadening and pulse nonselectivity. III. Uncertainty?

Csaba Szántay

Published Jul 1, 2008• 18 citations

Abstract

Abstract Following the treatments presented in Parts I and II, I herein discuss in more detail the popular notion that the frequency of a monochromatic RF pulse as well as that of a monochromatic FID is “in effect” uncertain due to the (Heisenberg) Uncertainty Principle, which also manifests itself in the fact that the FT‐spectrum of these temporal entities is spread over a nonzero frequency band. In Part III, I continue my preliminary review of some further fundamental concepts, such as the Heisenberg and Fourier Uncertainty Principles, that are needed to understand whether or not the NMR linewidth and the RF excitation bandwidth have anything to do with “uncertainty”. The article then culminates in re‐addressing our Two NMR Problems in a more conscientious frame of mind by using a more refined formalism. The correct interpretation of these problems will be discussed in Part IV. © 2008 Wiley Periodicals, Inc. Concepts Magn Reson Part A 32A: 302–325, 2008.

DOI: 10.1002/cmr.a.20116

Related papers

A Reformulative Retouch on the Fourier Transform – “Unprincipled” Uncertainty Principle

Csaba Szántay, Jr.

According to a widespread notion, a monochromatic harmonic temporal wave’s frequency ω becomes inherently “uncertain” to a degree ω Δ if the wave lasts for only a limited time t Δ owing to the “Uncertainty Principle” which states that constant ≥ ω Δ ⋅ Δt (i.e. ω Δ cannot be zero if t Δ is finite). This frequency uncertainty is commonly thought of as also being reflected in the fact that the (“sinc”-shaped) frequency spectrum obtained by Fouriertransforming a finite-duration monochromatic wave spreads over a nonzero frequency range. Since, as often argued, the Fourier spectrum represents the ”frequency components” of a temporal function (or signal), the nonzero width of the spectrum tells us that the ”nominally” monochromatic ω frequency of a time-limited sinusoid is “effectively” polychromatic, with frequencies continuously distributed over a ω Δ range in accord with the Uncertainty Principle.

InTech eBooksApr 11, 20121 citation

The Heisenberg Uncertainty Principle and Time-Frequency Decompositions

Barbara Burke Hubbard

It is sometimes said that one can't localize a signal simultaneously in time and frequency. This can be misleading. It's not that we can't localize the signal in time and frequency. The signal itself can't be concentrated simultaneously in time and in frequency. The Heisenberg uncertainty principle does not limit what we can know about reality; it describes that reality. The shorter-lived a function, the wider the band of frequencies given by its Fourier transform; the narrower the band of frequencies of its Fourier transform, the more the function is spread out in time.

A K Peters/CRC Press eBooksMay 30, 19980 citations

The uncertainty principle and resolution improvement

Dževad Belkić

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.

CRC Press eBooksAug 22, 20190 citations

Epitome Of Quantum Mechanics

M. Goldman

Abstract As stated in the preface, it is only through the use of quantum mechanics that one can get a real understanding of the working of the sophisticated methods that turned high-resolution NMR into a fundamental and powerful tool for the investigation of molecules, especially in the liquid state. However useful, the classical and naive quantum descriptions given in Chapter 1 suffice only for interpreting the simplest version of NMR spectroscopy, namely the resonance spectra obtained either by the CW method or by the Fourier transform of the FID following a single r.f. pulse. As regards methods such as 2D spectroscopy, these descriptions are fundamentally inadequate and provide at best a qualitative and more-or-less intuitive feeling of what happens, very often with serious conceptual difficulties.

Mar 28, 20010 citations

A physical model of Fourier uncertainty principle

Rohan Singh, Parbati Biswas

Abstract Fourier transform is an integral part of harmonic analysis, information theory and signal processing with wide applications. The transformation between the Fourier pair is governed by an uncertainty relation, most generally captured by the Bandwidth theorem, . The usual approaches to the uncertainty principle employ sophisticated methods which lack physical interpretation of the origin and description of the uncertainty for real systems. In this paper, we have proposed a physically inspired model of the Fourier transform with an aim to describe its underlying dynamics and how it leads to the fundamental uncertainty principle. A signal is modeled as helical spring in 3D space, which is wrapped around a circle with controllable wrapping frequency. This process is formulated to reproduce the Fourier transform. The model is used to describe the origin of uncertainty, and to formulate a new generator-based approach to the Bandwidth theorem.

Europhysics Letters (EPL)Jun 17, 20250 citations

The Heisenberg Uncertainty Principle and the Nyquist-Shannon Sampling Theorem

Pierre A. Millette

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

Jul 1, 20130 citations