Polynomial expansions are used to approximate the equations of the eigenvalues of the Schrödinger equation for a finite square potential well. The technique results in discrete, approximate eigenvalues which, it is shown, are identical to the corresponding eigenvalues of a wider, infinite well. The width of this infinite well is easy to calculate; indeed, the increase in width over that of the finite well is simply the original width divided by the well strength. The eigenfunctions of this wider, infinite well, which to first order has the same width for the ground state and all excited states, are also good approximations to the exact eigenfunctions of the finite well. These approximate eigenfunctions and eigenvalues are compared to accurate numeric calculations and to other approximations from the literature.
The photoionization cross sections for producing the O2+ parent ion and the O+ fragment ion from neutral O2 are presented from 650 to 120 Å. A new technique was used that eliminated the serious problem of identifying the true abundance of the O+ ions. These ions are generally formed with considerable kinetic energy and because most mass spectrometers discriminate against energetic ions true O+ abundances are difficult to obtain. In the present work the relative cross sections for producing O2+ are obtained and normalized against the total cross sections in a spectral region where dissociative ionization is not possible. The fragmentation cross sections for O+ were then obtained by subtraction of the O2+ cross section from the total photoionization cross section.
Higher-order cross and ordinary correlation detectors are applied to four deterministic transients contaminated by uncorrelated Gaussian noise only. Histograms and moments are used to examine the properties of the signals and their effect on detector performance. Receiver operating characteristic (ROC) curve analysis and limiting signal-to-noise ratios for ‘‘good’’ detection provide comparative measures for different detectors. Probability density functions of detection ordinate values of signal-present and noise-only correlations are used to explain ROC curve behavior. Using a known source, the cross-correlation detector performs better than the higher-order correlation detectors for each transient studied. However, for an unknown narrow pulse source signal, the bicorrelation and tricorrelation detectors outperform the cross-correlation detector. In contrast, the bicorrelation detector performs very poorly for low-frequency narrow-band signals with a small third moment embedded in uncorrelated Gaussian noise. Rectification as part of the detection process improves the performance of the bicorrelation detector and also places the peak of maximum magnitude at the origin. This eliminates the problem in detection or time delay estimation that the existence of multiple peaks due to symmetries in the auto-bicorrelation or the bicorrelation of repeated signals may cause. The tricorrelation detector also performs better with rectification than without. For an unknown source, the bicorrelation and tricorrelation detectors with rectification perform on a level comparable to the cross-correlation detector for certain signals. Comparisons are made between repeating a known source and repeating noisy received signals in the bicorrelation.
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