Quantum Mechanical Limits Research Articles

On Synchrotron Radiation Spectrum in Classical and Quantum Mechanical Limits

On Synchrotron Radiation Spectrum in Classical and Quantum Mechanical Limits

Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance

The quantum-mechanical framework in which observables are associated with Hermitian operators is too narrow to discuss measurements of such important physical quantities as elapsed time or harmonic-oscillator phase. We introduce a broader framework that allows us to derive quantum-mechanical limits on the precision to which a parameter—e.g., elapsed time—may be determined via arbitrary data analysis of arbitrary measurements onNidentically prepared quantum systems. The limits are expressed as generalized Mandelstam–Tamm uncertainty relations, which involve the operator that generates displacements of the parameter—e.g., the Hamiltonian operator in the case of elapsed time. This approach avoids entirely the problem of associating a Hermitian operator with the parameter. We illustrate the general formalism, first, with nonrelativistic uncertainty relations for spatial displacement and momentum, harmonic-oscillator phase and number of quanta, and time and energy and, second, with Lorentz-invariant uncertainty relations involving the displacement and Lorentz-rotation parameters of the Poincare group.

2H NMR studies for the examination of methyl group rotations between their classical and quantum mechanical limits

2H NMR spectra of copper acetate monohydrate [Cu(CD3CO2)2]·H2O, hexamethylbenzene-d18 and acetone-d6 were analyzed to characterize methyl group rotations between their classical and quantum mechanical limits. The temperature dependent spectra show the three different possible developments of the lineshapes. Hexamethylbenzene is typical of compounds with small but resolvable tunnelling frequencies. Characteristic splittings in the spectra of copper acetate monohydrate reveal the temperature dependence of tunnelling frequencies that are slightly larger than the quadrupole coupling constant. While the tunnelling frequency of acetone is already too large to be specified from2H NMR lineshape analysis at temperatures low enough that quantum effects influence the spectra, the temperature dependent spectra deviate characteristically from those found for classical rotation. Additionally to these experimental results, we outline how from an NMR point of view mixed classical and quantum rotations can be described by a single complex frequency.

"Coulomb logarithm" for inverse-bremsstrahlung laser absorption.

The ``Coulomb logarithm'' for inverse-bremsstrahlung laser absorption is examined for plasmas of different ionic charge, spanning the classical and quantum-mechanical limits. Previously, this term has not been calculated exactly for the conditions of interest in laser fusion experiments; it has only been estimated from physical considerations. For short-wavelength irradiation (e.g., 0.35 \ensuremath{\mu}m), uncertainties in the ``logarithmic'' factor can produce variations of 20\char21{}50 % in the laser absorption coefficient. A more exact treatment of this term is presented here. For low-Z plasmas, a modified Born approximation is used that reproduces previous results for long-range interactions that cannot be described by a single electron-ion collision, and it simultaneously treats the short-range electron-ion encounters. For high-Z plasmas, the Coulomb logarithm is calculated in terms of the classical, nonlinear electron trajectory in a self-consistent electrostatic potential; strong ion-ion correlations are treated by the nonlinear Debye-H\uckel model. There are no indeterminate quantities in the calculations.