NMR and the uncertainty principle: How to and how not to interpret homogeneous line broadening and pulse nonselectivity. I. The fundamentals

Abstract

Both the essence of homogeneous NMR line broadening as well as the fact that a short monochromatic radiofrequency pulse exhibits a polychromatic spectral bandwidth and is consequently associated with a broad off-resonance NMR excitation band are often rationalized in terms of a superficial and/or misguided use of the ''Uncertainty Principle.'' The problem is of quite fundamental nature, and stems from several deeper ''issues'' that reach beyond the world of NMR. To explore this topic, we need to go back to some basic concepts including the Fourier transform, time-frequency analysis, different versions of the ''uncertainty principle,'' and linear systems theory. My theme will be developed in four subsequent articles. First, by employing an approach that is unconventional in its thematic structure and applied nomenclature, I present an overview of some basic ideas (periodicity, basis functions, harmonic waves, density functions, etc.) that will then pave the way to discussing, in a somewhat avant-garde manner, the Fourier transform in Part II.

Carrier Density, Effective Mass, and Nuclear Relaxation Pathways in Plasmonic Sn:In2O3 Nanocrystals

“…The T 2 relaxation is not directly measured, but T 2 * can be extracted by relating the line width of the NMR signal to T 2 * where Δ ν fwhm is the fwhm of the NMR signal measured in hertz. The value of T 2 * is related to T 2 by the following expression where γΔ B 0 is the change in strength of the local magnetic field.…”

The RF Pulse and the Uncertainty Principle

“…The Heisenberg principle states that there is a minimum uncertainty in the simultaneous specification of [. This theme was already discussed in detail in a series of rather involved articles published previously, [6][7][8][9] which were subsequently also published in a shorter and more condensed form. .]…”

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

“…In a series of recent papers (Szántay, 2007(Szántay, , 2008a(Szántay, , 2008b(Szántay, , 2008c the author of the present work explored the above problem in detail, showing that a) the idea that the frequency of a time-limited sinusoid would be uncertain in the Heisenberg sense is flawed; b) it is the FUP (and not the HUP) that is of relevance to the problem; c) the notion that a monochromatic pulse "effectively" comprises of a range of physically existing frequency components is erroneous. During that discourse the need had come up to contemplate the essence of the Fourier Transform (FT) itself, which may appear somewhat odd considering the fact that the FT is ubiquitously applied in a broad range of sciences and technology, and works explaining its mathematics and its manifold physical applications are available in almost countless abundance.…”

“…From this formalism one can draw a number of subtle insights into the FT as was detailed in the original papers which also included a discussion on a geometrical interpretation of the FT and on its capability to unravel the oscillating frequency components that are otherwise obscured in a complicated time signal (Szántay, 2007(Szántay, , 2008a(Szántay, , 2008b(Szántay, , 2008c. These aspects will not be elaborated here.…”