Showing papers on "Dirac delta function published in 1980"

The inverse problem of electromagnetic induction: Existence and construction of solutions based on incomplete data

Abstract: A theory is described for the inversion of electromagnetic response data associated with one-dimensional electrically conducting media. The data are assumed to be in the form of a collection of (possibly imprecise) complex admittances determined at a finite number of frequencies. We first solve the forward problem for conductivity models in a space of functions large enough to include delta functions. Necessary and sufficient conditions are derived for the existence of solutions to the inverse problem in this space. The approach relies on a representation of real-part positive functions due to Cauer and an application of Sabatier's theory of constrained linear inversion. We find that delta-function models are fundamental to the problem. When existence of a solution has been established for a given set of data, actual conductivities fitting the measurements may be explicitly constructed for various special classes of functions. For a solution in delta functions or homogeneous layers a development as a continued fraction is the essential idea; smoothly varying models are found with an adaption of Weidelt's analytic solution.

Approximate solution of the Dirac equation using the Foldy-Wouthuysen hamiltonian

Abstract: The Foldy-Wouthuysen hamiltonian is used to obtain the energies of various states of the hydrogen-like atom. For the 1s and 2s states to order mc 2 α6 and for the 2p states to order mc 2 α8 divergent contributions to the energy arise, but it is shown that provided these divergences are handled properly they cancel one another and the correct finite result is obtained. An attempt to carry the calculations to higher orders using delta function recipes fails, but it proves possible to evaluate the mc 2 α8 energy contribution for the 1s state using a different method.

Weyl's association, Wigner's function and affine geometry

Abstract: According to Weyl one may associate a function with a dynamical operator; these functions depend on the parameters p and q and can be displayed in a p, q manifold, the W space. In the classical limit the W space becomes the phase space parametrised by the canonical variables. The function associated in this manner with the density operator is Wigner's function. It turns out that if Wigner's function is a delta function it cannot represent the density operator of a physically realisable state unless the argument of the delta-function is linear in the parameters a and q. In all other cases Wigner's function associated with a physically realisable state has a finite width, proportional to h 2 3 . Consequently straightness (linear combination of p and q) has a fundamental significance in the W space. Since this property is preserved under linear inhomogeneous transformations the W space will have a geometry generated by these transformations, the affine geometry of Euler, Moebius and Blaschke. In the present note we show how this comes about, how it simplifies the semiclassical approximations of Wigner's function, and makes one understand how in the classical limit this geometry is lost, allowing to be replaced by the geometry of canonical transformations.

Nodeless wave function quantum theory

Abstract: The nodes where the wave function changes sign in conventional quantum theory can alternatively be regarded as places where the wave function descends to zero but rises again without changing its sign. This behavior is accomplished by adding delta function barriers to the potential energy wherever a node occurs in the wave function. This model is supported by the recognition that the wave functions of a subsystem, which is not truly isolated, are really marginal amplitudes which implicitly average out interactions of the system with its environment. The finite effect of these interactions is to replace the delta function barriers by barriers of finite width and height.

On the application of the Sommerfeld–Maluzhinetz transformation to some one‐dimensional three‐particle problems

Abstract: The reformulation of the one‐dimensional three‐body problems with boundary condition and delta function interactions, based on the Sommerfeld–Maluzhinetz transformation, is presented. The argumentation is carried out as exemplified by two models–the exactly soluble model of two identical particles interacting through delta potential and each of which interacts with a third one through boundary condition interactions, and a model of two identical particles and a fixed wall, all interactions being of the delta function type. The problems are reduced to those of solving coupled systems of functional equations for the Sommerfeld transforms of the wavefunction. The functional properties of the transforms are then used to derive expressions relating them to the half off‐shell extensions of the elastic and exchange probability amplitudes as defined in the Faddeev–Lovelace approach.

Identification of distributed parameter systems via multidimensional distributions

Abstract: The paper presents a method of determining the parameters of a process described by a partial differential equation from a knowledge of its solution. The development is based on treating the process signals as multidimensional distributions in the manner established by Laurent Schwartz, and expanding them in an exponentially weighted series of the generalised partial derivatives of the multidimensional Dirac delta function, termed as the Poisson moment functional ( p. m. f. ) expansion. The ability of the method is successfully demonstrated in the presence of noise.

Simple exact solution of the one-dimensional Schroedinger equation with continuous plus delta-function potentials of arbitrary position and strength

Abstract: A simple exact solution of a one-dimensional Schroedinger equation which describes the motion of a particle in a continuous potential V(x) and scattering from a finite set of delta-function potentials of arbitrary positions and strengths is found.

Imaginary part of the ion-ion interaction potential

Abstract: The energy density formalism of Brueckner is extended to the complex domain with the use of a complex two-body effective interaction, so as to generate simultaneously the real and imaginary part of the ion-ion interaction potential. A simple effective interaction consisting of a density-dependent delta function repulsion and a Gaussian attraction is chosen for the real part of the complex effective interaction, whereas the imaginary component is derived by using the forward scattering amplitude approximation. In using this approximation, emphasis has been given to use the complex two-body matrix T, where the propagator has the nuclear many-body Hamiltonian rather than the free two-body matrix t. For the average nucleon-nucleon cross section , required in constructing the complex effective interaction, a modified version of the relation suggested by Bohr and Mottelson is used which compares quite well with that of Clemental and Villi. The real part of the ion-ion interaction potential calculated in this model has already been reported in an earlier work and the present work deals with the calculation of the imaginary part of the potential for five pairs of spherical nuclei. The region of the potential beyond the surface is parametrized into a Woods-Saxon shape. The calculated imaginarymore » potential for the system /sup 40/Ca-/sup 16/O is compared with the phenomenological one of Becchetti et al. and the agreement between the two sets of potentials is found to be quite good in the region beyond the strong absorption radius.« less

Coordinate measurements and operators

Abstract: Relativistic quantum-field theory provides the machinery for calculating wave functions or probability amplitudes depending upon space-time coordinates. The currently accepted theory, however, fails to provide position operators and a means of measuring particle coordinates that are consistent with Dirac's properties of physical observables. This is because it calls for a space position probability distribution at a specified time. This paper shows, however, that space-time event coordinate operators, together with a corresponding measurement procedure, can be found that are consistent with Dirac's requirements. This is done through a reinterpretation of the amplitudes computed by field theory and does not involve any change in that mathematical formalism. The measurement of the space-time coordinates of an event is accomplished by detecting the absorption of a photon by a particle from each of two light pulses designed to overlap at a given point at a given time. If a final emitted photon has an energy whose sum with the final particle energy approximately equals the sum of the mean energies of the pulses, then the absorption of the two pulse photons must certainly have taken place within a distance the order of a Compton wavelength of the small space-time region of overlapping pulses. This is clear from the fact that the high energy required to confine the pulses to very small volumes must throw a particle absorbing them far off the mass shell. Thus the absorption of the two photons throws the particle into a narrowly confined spatial wave function that must decay extremely rapidly—to within a Compton wavelength, a delta function in space-time. This delta function is the eigenfunction of space-time coordinate operators Xμ and is the scalar product of vectors in a Hilbert space spanned by spin–space-time kets large enough to contain the operators of the Poincare group. These event operators transform properly under the action of Poincare operators but do not commute with the mass. If the Compton wavelength is not negligible compared to the accuracy desired in the coordinate measurements, individual coordinate measurements are no longer possible. Nevertheless, a large number of repeated coordinate measurements can be carried out to produce a coordinate probability distribution. This distribution can be unfolded to find a true coordinate probability distribution if the charge form factor is known from basic theory. An analysis of laboratory particle detection techniques shows that they actually determine space coordinates and energy rather than spatial coordinates at a given time. When this fact is included, the Klein–Nishina formula can be derived using the electromagnetic four-vector potential as the photon probability amplitude wave. To clarify the meaning of the observables, a mass-momentum measurement is described.

Solutions for point loads and generalized circular loads applied within gross anisotropic media

Abstract: This paper presents solutions to the full range of load types applied to circular areas located within cross anisotropic elastic media. Point loads are considered as a special case. The five load types are: (i) vertical force, (ii) horizontal force, (iii) moment about a horizontal axis, (iv) moment about a vertical axis, (v) radial shear stresses. For any particular problem the elastic medium may be in the form of a half space or a full space. In both cases integral transform techniques are used to derive general solutions for all displacement and strain components for quite general forms of distribution of the loading stresses. The particular loading stress distributions of the forms (1-r**2)**q and r(1-r**2)**t are of practical importance and the solutions in these cases can be expressed as sums of hypergeometric functions. The solutions for point loads are relatively simple functions produced by introducing the dirac delta function. The solutions produced have practical importance in soil and rock engineering for conditions where there are subsurface loadings and/or discontinuities, either natural or man made (a).

Influence of the self-energy diagrams on the solutions of a scalar Bethe-Salpeter equation

Abstract: The influence of self-energy diagrams on solutions of the Bethe-Salpeter equation is studied by taking a gphi*//sub 1/(x)phi/sub 1/(x)phi/sub 2/(x) interaction in the ladder approximation The results show that for ground-state solutions the self-energy diagrams will diminish the eigenvalues and alter the wave functions slightly However, for the excited states and antisymmetric solutions, the influence of the self-energy diagrams is considerable, and completely alters the properties of the solutions Their wave functions look like a delta function and their eigenvalues are independent of the binding energy and the quantum numbers The results also show that the infrared behavior of the self-energy diagrams is very important for the solutions of the equation, and that the solutions from the ladder approximation are meaningful only in the loosely bound cases As the binding becomes tight, the contribution from the self-energy diagrams becomes important