Lifetime Broadening of a Nuclear Magnetic Resonance Peak under Minimal Length Uncertainty Analysis

Abstract

It is expected that some low energy relics of high energy phenomena may be verified by nowadays experiments, and some phenomenological low energy effects may be explored in solvable models at a level of quantum mechanics in a minimal length scale [1–9]. In literature, the minimal length quantum mechanics have been extensively studied [10, 1, 2]. In this letter we study the possibility of testing minimal length scale via nuclear magnetic resonance spectroscopy taking into account the solution of broadening of a magnetic resonance peak by minimal length uncertainty. One manifestation of the uncertainty principle is the effect of decreasing the lifetime of a nucleus in a given environment on the nuclear magnetic resonance spectrum of the same nucleus or one to which it is coupled. These kinetic effects may have experimental significance by giving rise to spectra whose shift-state values or line widths may be used as a measure, or to spectra which are simplified so that the resonances are more suitable for quantitative measurements. The shifting and merging of spectra by kinetic process may also be used to shift an increasing resonance away from a resonance of interest if the interfering species can be made to undergo rapid exchange. The magnitude of the effect can be estimated from uncertainty principle, ∆E∆t ≥ ~. Since ∆E = ~∆ν and ∆t may be identified with τ , the average lifetime of the excited state in the various exchanging states, the relationship becomes τ∆ν ≈ 1/2π. Thus the magnitude of the rate constant that will give an observable effect will vary for different types of spectrometry, depending on the frequency difference involved. The widest application of this principle has been in nuclear magnetic resonance spectrometry, especially proton NMR. When a system