Showing papers on "Dirac delta function published in 1982"

A unified and consistent view on the singularities of the electric dyadic Green's function in the source region

Abstract: Ambiguities associated with the singularity of the electric dyadic Green's function are clarified. The numerical advantages of finite and infinitesimal principal volumes are dependent on the particular problem being treated. It is shown that apparent discrepancies are nonexistent and that either the method of potential or the theory of distribution can be used in this problem and a unified and consistent view can be formed.

Products and compositions with the Dirac delta function

Abstract: The need to define pointwise products and compositions with distributions is pointed out in the context of the problems of renormalisation, junction conditions and curved shock waves. Earlier definitions are briefly reviewed, and new definitions are proposed using non-standard analysis. Basic properties are established, and some products and compositions with the delta distribution are explicitly evaluated. With these definitions, the domain of validity of the nonlinear differential equations of classical field theory can be extended to include Rankine-Hugoniot equations are derived from the Euler equations. An immediate application to quantum field theory is pointed out.

Localization in an almost periodically modulated array of potential barriers

Abstract: It is argued that an array of potential barriers modulated in position and strength in a manner incommensurate with the barrier spacing can be replaced by a $\ensuremath{\delta}$-function Kronig-Penny model. With this equivalence it is shown, with use of renormalization-group arguments as well as known results from other calculations, that this model possesses both localized and extended states separated by mobility edges.

Non-resonant effects in the fluorescent Dicke model. II. Semiclassical and quantal results

Abstract: For pt.I see ibid., vol.14, p.4171 (1981). Semiclassical and quantal time-dependent results are presented for a model of N identical two-level atoms occupying the same site (the Dicke model) and driven by a CW off-resonance laser field. The exact semiclassical time-dependent solutions are derived from an analysis of the Fokker-Planck equation for the system in the atomic coherent states representation. Steady-state behaviour predicted by these solutions is in agreement with that obtained from the exact quantal steady-state analysis in the thermodynamic limit N to infinity . The semiclassical theory predicts (in the steady state) one type of fluorescent spectrum, which is a single sharp line ( delta function) centered at the driving field frequency. This is like the case at exact resonance and below threshold. Quantum mechanically, analytical results are obtained in the strong-field limit and within the secular approximation of the master equation: the fluorescence spectrum is the usual dynamical Stark triplet, apart from a cooperative factor N2. The absorption spectrum and the second-order intensity autocorrelation function are also calculated. Numerical results are also presented for finite N(N

On quantisation using periodic classical orbits

Abstract: Two approaches to semiclassical quantisation of integrable systems using periodic classical orbits are considered. They both lead to approximate formulae for the density of states function (a delta function at each energy level). The first, due to Gutzwiller (1971), involves a sum over isolated stable periodic orbits of the system, and leads to the harmonic approximation to the eigenvalues. The second, due to Berry and Tabor (1976), involves a sum over families of periodic orbits, and leads to the EBK ('torus') approximation to the eigenvalues. Here, the author extracts a modified form of the Gutzwiller series from the Berry-Tabor series by using a uniform approximation, and hence show that the complete spectrum involves both these series. The analysis demonstrates that genuine semiclassical quantisation rules for generic systems, using periodic orbits, will involve uniform approximation, which more closely reflects the underlying classical structure than do the existing stationary phase approximations.

A linear stochastic model of the single motor unit

Abstract: The production of force and of the electrical signal by an active motor unit is theoretically described. Neural spikes are modelled using the Dirac delta function. Mechanisms for the generation of random impulse trains and the properties of the corresponding stochastic processes are discussed; the "renewal" model is proposed as the most appropriate. The possibility of using a linear model for the systems that produce force and electrical signal in the unit is examined. It is concluded that the linear assumption is justifiable during steady, constant-strength contractions of muscle. This linear stochastic model of the motor unit is used in two subsequent papers to study the muscle force and the electromyogram.

Nonclassical fields with singularities on a moving surface

Abstract: Fields with singularities on a moving surface S with boundary ∂S can be represented as distributions which have their support concentrated on S and ∂S. This paper considers such fields of the form F={ f }+λδS, where { f } is the distribution determined by a field f and λδS is a Dirac delta distribution with density λ concentrated on the tube S swept out by the moving surface. A straightforward calculation of the distributional gradient, curl, divergence, and time derivative of such fields yields fields of the following general form: G={ g } +αδS +βδ∂S +γ∇n(⋅)δS. The density α is shown to contain all the information which is customarily presented in the jump conditions for fields with singularities at a moving interface. Examples from electromagnetic field theory are presented to show the significance of the other terms { g }, βδ∂S, and γ∇n(⋅)δS.

One‐dimensional hydrogen atom in an infinite square well

Abstract: The effect of boundary conditions on solutions to the Schrodinger Equation is demonstrated.

Discontinuity in the first derivative of the Schrödinger wave function

Abstract: Bound state problems in quantum mechanics are considered for the edge of a square well potential and singular points of a Coulomb and a δ‐function potential. (AIP)

Validity of a selective summation technique in analysis of wave propagation through random media

Abstract: The so-called moment equation for a successively forward scattered wave was derived under a definite condition. The condition shows that the applicability of the equation is much more extensive. When a selective summation technique is applied to the derivation of the moment equation, another condition besides the above is necessary for the existence of the equation. The extra condition is due to the selective summation of scattered waves and shows the validity of the technique for the nth moment of forward scattered waves. For example, if the waves propagate along the z axis, a necessary condition for the technique to be valid for the first, second, and fourth moments is k2l2B(0, 0) ≪ 1 where k is the wave number in free space and l and B(r, z) are the correlation length and the correlation function of fluctuation of the medium, respectively. For all the higher moments the technique is applicable only to the case of B(r, z) = Br,(r)δ(z) where δ(z) is the Dirac delta function.

Repulsive Delta-Function Potential in Two and Three Dimensions

Abstract: It is shown by choosing an appropriate variational wave function that the ground state energy is not affected by the repulsive delta-function potential in two and three dimensional systems. Scattering cross section by the repulsive delta-function potential is shown to be zero.

Superreals and superfunctions

Abstract: The system of superreal numbers introduced by D. O. Tall shows that it is possible to develop a limited theory of infinitesimals (and infinite numbers) in an elementary context, using only classical mathematical ideas and methods. It is used here to present a simple and intuitively straightforward introduction to the delta function and its derivatives. The treatment has the conceptual advantage of exhibiting the delta function as a genuine function in some sense, while at the same time making it clear how that sense differs from that appropriate to ‘ordinary’ functions.

Heat conductivity in a moderately dense hard sphere gas

Abstract: The three contributions to the heat conductivity in a moderately dense hard sphere gas are calculated using the Green–Kubo expressions for the kinetic, the cross, and the potential part of the time‐correlation function. The starting point is the calculation of the relevant correlation functions within the ’’successive uncorrelated binary collision’’ model. It is shown that the cross‐correlation function is proportional to the kinetic correlation function and that the potential correlation function contains two contributions, one proportional to the kinetic correlation function and the other proportional to a Delta function. This latter part is due to the instantaneous nature of the momentum and energy transfer in a hard sphere collision. The time integrals over these correlation functions provide the three contributions to the heat conductivity coefficient.