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On Dirac equations for linear magnetoacoustic waves propagating in an isothermal atmosphere
A new analytical approach to study linear magnetoacoustic waves propagating in an isothermal, stratified, and uniformly magnetized atmosphere is presented. The approach is based on Dirac equations, and the theory of Sturm-Liouville operators is used to investigate spectral properties of the obtained Dirac Hamiltonians. Two cases are considered: (1) the background magnetic field is vertical, and the waves are separated into purely magnetic (transverse) and purely acoustic (longitudinal) modes; and (2) the field is tilted with respect to the vertical direction and the magnetic and acoustic modes become coupled giving magnetoacoustic waves. For the first case, the Dirac Hamiltonian possesses either a discrete spectrum, which corresponds to standing magnetic waves, or a continuous spectrum, which can be clearly identified with freely propagating acoustic waves. For the second case, the quantum mechanical perturbation calculus is used to study coupling and energy exchange between the magnetic and acoustic components of magnetoacoustic waves. It is shown that this coupling may efficiently prevent trapping of magnetoacoustic waves instellar atmospheres.
An Exact Separation of the Spin-Free and Spin-Dependent Terms of the Dirac-Coulomb-Breit Hamiltonian
The Dirac Hamiltonian is transformed by extracting the operator (sigma x p)/2mc from the small component of the wave function and applying it to the operators of the original Hamiltonian. The resultant operators contain products of Paull matrices that can be rearranged to give spin-free and spin-dependent operators. These operators are the ones encountered in the Breit-Pauli Hamiltonian, as well as some of higher order in alpha(sup 2). However, since the transformation of the original Dirac Hamiltonian is exact, the new Hamiltonian can be used in variational calculations, with or without the spin-dependent terms. The new small component functions have the same symmetry properties as the large component. Use of only the spin-free terms of the new Hamiltonian permits the same factorization over spin variables as in nonrelativistic theory, and therefore all the post-Self-Consistent Field (SCF) machinery of nonrelativistic calculations can be applied. However, the single-particle functions are two-component orbitals having a large and small component, and the SCF methods must be modified accordingly. Numerical examples are presented, and comparisons are made with the spin-free second-order Douglas-Kroll transformed Hamiltonian of Hess.
A method for calculation of free-space sound pressures near a propeller in flight including considerations of the chordwise blade loading
This report presents tabulated values of certain definite integral that are involved in the calculation of near-field propeller noise when the chordwise forces are assumed to be either uniform or of a Dirac delta type. The tabulations are over a wide range of operating conditions and are useful for estimating propeller noise when either the concept of an effective radius or radial distributions of forces are considered. Use of the tabulations is illustrated by several examples of calculated results for some specific propellers.
The Hamiltonian structure of Dirac's equation in tensor form and its Fermi quantization
Currently, there is some interest in studying the tensor forms of the Dirac equation to elucidate the possibility of the constrained tensor fields admitting Fermi quantization. We demonstrate that the bispinor and tensor Hamiltonian systems have equivalent Fermi quantizations. Although the tensor Hamiltonian system is noncanonical, representing the tensor Poisson brackets as commutators for the Heisenberg operators directly leads to Fermi quantization without the use of bispinors.
Quantum principles and free particles
The quantum principles that establish the energy levels and degeneracies needed to evaluate the partition functions are explored. The uncertainty principle is associated with the dual wave-particle nature of the model used to describe quantized gas particles. The Schroedinger wave equation is presented as a generalization of Maxwell's wave equation; the former applies to all particles while the Maxwell equation applies to the special case of photon particles. The size of the quantum cell in phase space and the representation of momentum as a space derivative operator follow from the uncertainty principle. A consequence of this is that steady-state problems that are space-time dependent for the classical model become only space dependent for the quantum model and are often easier to solve. The partition function is derived for quantized free particles and, at normal conditions, the result is the same as that given by the classical phase integral. The quantum corrections that occur at very low temperatures or high densities are derived. These corrections for the Einstein-Bose gas qualitatively describe the condensation effects that occur in liquid helium, but are unimportant for most practical purposes otherwise. However, the corrections for the Fermi-Dirac gas are important because they quantitatively describe the behavior of high-density conduction electron gases in metals and explain the zero point energy and low specific heat exhibited in this case.
Curved Focal-Plane Arrays Using Back-Illuminated High-Purity Photodetectors
Curved-focal-plane arrays of back-illuminated silicon-based photodetectors are being developed. The basic idea is to improve the performance of an imaging instrument and simplify the optics needed to obtain a given level of performance by making an image sensor (e.g., a photographic film or an array of photodetectors) conform to a curved focal surface, instead of following the customary practice of designing the optics to project an image onto a flat focal surface. Eyes are natural examples of optical systems that have curved focal surfaces on which image sensors (retinas) are located. One prior approach to implementation of this concept involves the use of curved-input-surface microchannel plates as arrays of photodetectors. In comparison with microchannel plates, these curved-focal-plane arrays would weigh less, operate at much lower voltages, and consume less power. It should also be possible to fabricate the proposed devices at lower cost. It would be possible to fabricate an array of photodetectors and readout circuitry in the form of a very-large-scale integrated (VLSI) circuit on a curved focal surface, but it would be difficult and expensive to do so. In a simple and inexpensive alternate approach, a device (see figure) would have (1) a curved back surface, onto which light would be focused; and (2) a flat front surface, on which VLSI circuitry would be fabricated by techniques that are well established for flat surfaces. The device would be made from ultrapure silicon, in which it is possible to form high-resistivity, thick photodetectors that are fully depleted through their thicknesses. (As used here, "thick means having a thickness between a fraction of a millimeter and a few millimeters.) The back surface would be polished to the curvature of the focal surface of the intended application. To enable the collection of charge carriers excited by photons near the back surface or in the bulk of the device, it would be necessary to form a transparent or semitransparent back-surface electrode, possibly by delta doping. [Delta doping is so named because its density-vs.-depth characteristic is reminiscent of the Dirac function (impulse function): the dopant is concentrated in a very thin layer - nominally, a single atomic layer.]