Nonzero Imaginary Part Research Articles

Two critical field strengths in the Copenhagen vacuum

One-loop corrections to the equation expressing the instability of a constant colour magnetic field with respect to the Nielsen-Olesen mode are considered. These corrections radically change the equation, and introduce a critical field strength B C upon which instability, and consequently the life or death of the Copenhagen vacuum, depends. The value of B C is calculated, and is found to leave the Copenhagen vacuum intact. Our equation distinguishes between field configurations, which are quantum dynamically unstable and configurations having an effective energy with non-zero imaginary part, generally believed to signal instability. We interpret the difference. Finally we expose a mechanism which presumably makes the imaginary part of the effective energy density vanish, if the Nielsen-Olesen unstable mode is excited above a critical amplitude χ C ≈ 0.24.

Some mathematically simple ensembles of random matrices which represent Hamiltonians with a small time-reversal-noninvariant part

Some of the statistical properties corresponding to Gaussian ensembles of random matrices whose matrix elements are not all functionally independent are investigated. Three types of ensembles are studied. First, an ensemble whose matrix elements are all real is examined. Next, an ensemble whose matrix elements are not all real, but whose off-diagonal elements have real and imaginary parts which are of the same size on the average is studied. Finally, ensembles with of order-N and-N2 nonzero imaginary parts of arbitrary size are investigated. Each of the above ensembles is compared with the corresponding Gaussian ensemble in which all of the nonzero matrix elements are functionally independent.

SCF treatment of charge polarization effects in intermediate-energy electron scattering calculations with applications to N2

We report converged rotational close coupling calculations of the differential, integral, and momentum-transfer cross sections for seven model potentials for electron–N2 scattering at an impact energy of 30 eV. The model potentials involve a static potential calculated by the INDO/1s or INDOXI/1s method, an exchange potential calculated by the semiclassical exchange approximation from the INDO/1s or INDOXI/1s unperturbed electronic density, and a polarization potential. The polarization potentials used include the Buckley–Burke semiempirical one and various modifications of the INDOXI and INDO self-consistent-field adiabatic polarization potentials. We are able, without adjusting parameters, to obtain good agreement with the angle dependence of the experimentally measured sum of the elastic and rotational excitation differential cross sections although the absolute value of our calculated cross sections is about 20%–30% larger than the measured values in the best case, perhaps indicating that the model potentials are too strong or should have a nonzero imaginary part. We discuss the sensitivity of the computed results to details of both the static and polarization parts of the model potentials, and we present some predictions of the rotationally resolved state-to-state cross sections.