Wigner Formula Research Articles

Calculation of resonances in the Coulomb three-body system with two disintegration channels in the adiabatic hyperspherical approach

The method of calculation of the resonance characteristics is developed for the metastable states of the Coulomb three-body system with two disintegration channels. It is based on the numerical solution of the scattering problem in the framework of the adiabatic hyperspherical approach. The energy dependence of K-matrix in the resonance region is calculated with the use of the stabilization method. The resonance position E0 and the partial widths Γ1, Γ2 are obtained by fitting of the numerically calculated Kij(E) with the help of the general Breit–Wigner formula which takes into account the non-zero background inelastic scattering. The method developed is applied to the calculation of the parameters of the three lowest metastable states of the mesic molecular ion dt μ.

Charging effects in biased molecular devices

The influence of charging effects on the transport characteristics of a molecular wire bridging two metallic electrodes in the limit of weak contacts is studied by the generalized Breit–Wigner formula. Molecule is modeled as a quantum dot with discrete energy levels, while the coupling to the electrodes is treated within a broad-band theory. Owing to this model we find self-consistent occupation of particular energy levels and orbital energies of the wire in the presence of transport. The nonlinear conductance and current–voltage characteristics are investigated as a function of bias voltage in the case of symmetric and asymmetric coupling to the electrodes. It is shown that the shape of those curves are determined by the combined effect of the electronic structure of the molecule and by electron–electron repulsion.

Resonance scattering on In impurity in Pb 0.4Sn 0.6Te solid solutions

The Hall coefficient and resistivity were measured on a series of samples of InTe in Pb 0.4Sn 0.6Te solid solutions with varying indium content from 0 to 5 molar% . All samples show a p-type conductivity with the concentration of holes ∼10 19 cm −3 at 77 K . A considerable decrease of mobility calculated from the measurements was observed in the samples with high content of In. The contribution to the mobility, directly related to the scattering of holes into In atoms, was determined, and the scattering crosssection was estimated. We observed a drastic increase in the scattering crosssection with the rise of In content, which can be explained by the resonance scattering of holes into the band of quasi-local states of In. This result can also be explained with the use of the Breit–Wigner formula. The location of the In states in the valence band was also calculated using the Kane model.

Breit–Wigner formula at barrier tops

For noncritical energies, the asymptotic behavior of the scattering phase and of the time-delay are known to be described by a Weyl type formula and the Breit–Wigner formula, respectively. We consider here the case of critical energy levels in dimension 1. We obtain the semiclassical asymptotics of the scattering phase and of the time-delay, uniformly with respect to the energy in a neighborhood of a critical value.

A new renormalization scheme for the fermion field in the standard model

In order to obtain proper wavefunction renormalization constants for unstable fermions and be consistent with Breit–Wigner formula in the resonant region, we have assumed an extension of the LSZ reduction formula for unstable fermions and adopted an on-shell mass renormalization scheme in the framework without field renormalization. The comparison of the gauge dependence of physical amplitude between on-shell mass renormalization and complex-pole mass renormalization has been discussed. After obtaining the fermion wavefunction renormalization constants, we extend them to two matrices in order to include the contributions of off-diagonal two-point diagrams at fermion external legs for the convenience of calculations of S-matrix elements.

Modified Breit–Wigner formula for mesonic resonances describing OZI decays of confined $q\bar{q}$ states and the light scalar mesons

A general expression resembling the Breit–Wigner formula is derived for the description of resonances which appear in meson–meson scattering. Our starting point is a unitarised meson model, but reduced to a simpler form and freed from the specific assumption about the confining force. The parameters of the resulting “resonance-spectrum expansion” are directly related to the confinement spectrum and the mechanism of $^3P_0$ valence-quark-pair creation for OZI-allowed hadronic decay, and not to the central positions and widths of resonances. This method also provides us with a straightforward explanation for the origin of the light scalar mesons without requiring extra degrees of freedom.

The Greuling–Goertzel and Wigner approximations in the theory of neutron slowing down

The classic Greuling–Goertzel and Wigner approximations for neutron slowing-down problems are reappraised theoretically and numerically. We sketch the standard derivations of these approximations and show that, contrary to conventional wisdom, their theoretical bases are comparable. We also present a new derivation of these approximations, which clarifies the relationship between them and the Fermi age approximation. Finally, we compare numerical solutions of the exact, Greuling–Goertzel, and Wigner equations. These results show that for narrow absorption resonances described by the Breit–Wigner formula, the Wigner solution is significantly more accurate than the Greuling–Goertzel solution.

<I>Ab Initio</I> Calculation on the Structure and Elastic Properties of a Magnesium-Lithium Alloy

Lattice constants, bulk modulus and heat of solution of a Mg 7 Li 1 alloy are calculated by using an ab initio pseudopotential method based on the density function theory and the supercell approach. Lattice constants and bulk modulus at zero pressure are obtained by using the stress calculation and generalized Hooke's law, For exchange and correlation method, the Wigner formula are used and the results are compared with ones using the Perdew and Zunger formula. Calculated lattice constants and bulk modulus are in good agreements with experimental values. We also calculate three kinds of ordered structures on this alloy in order to examine the influence of the structure. The results of three structures are different, respectively, and the influence is discussed from the atomic and electronic structure. The valence charge distribution is decreased around Li atoms. Especially, this region of the low charge distribution spreads in c direction.

Eugene Paul Wigner. 17 November 1902 — 1 January 1995

Eugene Wigner was a towering leader of modern physics for more than half of the twentieth century. Although his greatest renown was associated with the research–article of symmetry theory to quantum physics and chemistry, for which he was awarded the Nobel Prize in Physics for 1963, his scientific work encompassed an astonishing breadth of science, perhaps unparalleled during his time. In preparing this memoir, we have the impression we are attempting to record the monumental achievements of half a dozen scientists. There is the Wigner who demonstrated that symmetry principles are of great importance in quantum mechanics; who pioneered the application of quantum mechanics in the fields of chemical kinetics and the theory of solids; who was the first nuclear engineer; who formulated many of the most basic ideas in nuclear physics and nuclear chemistry; who was the prophet of quantum chaos; who served as a mathematician and philosopher of science; and the Wigner who was the supervisor and mentor of more than forty PhD students in theoretical physics during his career of over four decades at Princeton University. The legacy of these contributions exists in two forms. First, there are the papers–in excess of 500–now included in eight volumes of his collected works (15)*. His legacy also resides in the many concepts and phenomena that bear his name. There is, for example, the Wigner–Eckart theorem for the addition of angular momenta, the Wigner effect in nuclear reactors, the Wigner correlation energy, as well as the Wigner crystal in solids, the Wigner force, the Breit–Wigner formula in nuclear physics, and the Wigner distribution in the quantum theory of chaos. is collection of essays Symmetries and reflections (14) provides an insightful view of the many intellectual matters that concerned him during a busy career. The recollections of his life recorded by Andrew Szanton when Wigner was in his eighties (Szanton 1992) provide a special insight into the circumstances of his life and the incidents that brought him to the fore.

VisRes: A G RACE tool for displaying and analysing resonances

VisRes is a graphical tool for displaying and analysing resonance data. Files of eigenphase sums, generated during the computation of electron and photon interactions with atoms and molecules, can be read and the data displayed both as continuous and discontinuous graphs. Resonance resolution can be improved by interactively computing additional data points and merging them with the displayed data. Resonance positions and widths are determined by fitting user selected data points by the Breit—Wigner formula. VisRes is written in C and Motif for the UNIX operating system. It can be operated exclusively on the host workstation or in distributed mode where data is read from and computed on a remote computer but displayed and analysed on the host workstation. While it is published as a stand-alone tool, VisRes also forms part of the Graphical R-matrix Atomic Collision Environment (G RACE).

Matrix theory of elastic resonance scattering

A fundamental matrix theory is developed for the exact isolation (in both magnitude and phase) of resonance amplitudes contained in the elastic field scattered by a penetrable target. Similarities and differences between this matrix theory and the classical resonance scattering theory [L. Flax, L. R. Dragonette, and H. Uberall, J. Acoust. Soc. Am. 63, 723–731 (1978)] are discussed. The matrix theory is then applied to cylindrical and spherical fluid-filled cavities, and analytical expressions for the resonance amplitudes are derived. In the vicinity of resonance frequencies, the resonance amplitude expressions have forms similar to the Breit–Wigner formula. In order to check the validity of the matrix theory, numerical calculations are performed for a cylindrical water-filled cavity in an aluminum medium. Resonance spectra of the cavity are presented for some partial waves and their characteristics are also discussed.

Edge Effects and Scattering in Resonant Tunneling Electron Devices

The electron transport mechanisms and the current-voltage characteristics in one- and two-dimensional structures are investigated in double-barrier heterostructure electron devices. The resonant tunneling characteristics due to the reduced size of semiconductor devices are theoretically studied. We compare our results with calculations obtained for the case of weak lateral confinement (3 dimensional to 2 dimensional tunneling when the gated resonant tunneling diode (GRTD) becomes two-dimensional in the well region). Using the Breit and Wigner formula in the case of weak lateral confinement, the elastic and inelastic scattering mechanism are studied. As scattering time τi decreases, the current-voltage characteristics substantially broaden and the peak-to-valley current ratio decreases. We observe the edge effect and negative transconductance in the current-voltage characteristic of the gated resonant tunneling diode.

The effect of interaction between supported Au clusters on the conductance resonance

Conductance resonance of tunneling structure with a few Au clusters, deposited on an insulating film, is studied by the generalized Breit–Wigner formula in a tight-binding approximation. We find that in the conductance resonance the multiple peak structure may come from the interaction between supported Au clusters on an insulating film and the different arrangements of Au clusters may cause difference of the conductance resonance peaks. The experimental observation of multiple peak structure can be explained by generalizing our model to the system of a few quantum dots. Therefore, it is possible to predict the effect of interaction between metal clusters on the conductance resonance and develop some new microelectronic devices by artificially arranging metal clusters onto the surface of the insulating film.

Conductance resonance of metal-insulator-metal junction with embedded metal cluster

Conductance resonance of a metal-insulator-metal junction with an embedded metal cluster, which has different size and structure, is studied by the generalized Breit–Wigner formula in a tight-binding approximation. We find that a certain metal cluster possesses a specific peak pattern in the conductance data, that is, the conductance resonance peaks depend on the size and structure of metal clusters and the resonant broadening Γ is related to the tunnel barrier parameters. This arrangement can induce a high-quality tunneling structure with very narrow resonance peaks. Therefore, both the size effect and structure effect of the metal clusters can be predicted by observing conductance resonance of tunneling structure of embedding metal clusters in metal-insulator-metal junction. It is also possible to develop some new microelectronic devices with this structure.

Canonical approximate quantum measurements

In order to generalize the Wigner formula for the joint probability distribution of successive canonical measurements of discrete observables to continuous observables, a mathematical model is investigated of a canonical approximate measurement of an arbitrary observable, which generalizes von Neumann’s model of measuring interaction and Davies’s covariant instrument for approximate position measurement. Calculations of two types of root-mean-square errors show that this model clears certain requirements for good approximate measurements. A modified Wigner formula for successive canonical approximate measurements of arbitrary observables is established. It is also shown that the canonical approximate measurement of a discrete observable is reduced to the conventional measurement satisfying the von Neumann–Lüders collapse postulate by a minor modification in the process of measurement.

Relativistic interactions by means of boundary conditions: The Breit–Wigner formula

The relativistic generalization of the Breit–Wigner formula presented in this paper is not based on perturbative quantum field theory. Rather, it starts with two free Klein–Gordon or Dirac particles interacting only at the point of coincidence in space-time. This state has then two components, the two-particle one just mentioned and a single-particle one representing the compound system. From considerations of conservation of probability the boundary condition is derived relating the two-particle component at coincidence with the one-particle component of this state. From this boundary condition, the R and S matrices of the problems are obtained as well as a Breit–Wigner type of cross section. The present paper is restricted to a scalar compound particle so its spin is 0 and its parity +. In the concluding section, the generalization to one-point, but still Poincaré invariant, boundary conditions is mentioned. This generalization allows one to go to compound particles with arbitrary spin and parity and, in particular to the case 1− which may be of some practical interest.

Effects of inelastic electron-phonon scattering on the resonant tunneling in double-barrier structure

The Breit–Wigner formula is exploited to calculate the tunneling transmission probability affected by longitudinal optical phonon scattering in multiple quantum wells composed of Ga1−xAlxAs compounds. A quantitative result shows that the resonance electron energy and the integrated total transmission probability through the potential barriers are not changed, while the transmission peak is reduced and the linewidth is broadened.

Correlation correction to the Hartree-Fock total energy of solids. II

A posteriori corrections to the Hartree-Fock total energy of crystalline compounds are obtained with the density functional proposed by Colle and Salvetti [Theor. Chem. Acta 37, 329 (1975)]. Results are presented for diamond, graphite and metallic lithium. For covalent systems, the error on the atomization energy is of the order of 8%, both with small and large basis sets. For metallic lithium the correlation energy is more severely underestimated but the error can be reduced to a few percent using a mixed scheme, which utilizes the CS functional within the ionic sphere, and the Wigner or Ceperley-Alder formula in the interstitial region.

A general law for quantum mechanical joint probabilities: generalisation of the Wigner formula and the collapse postulate for successive measurements of discrete as well as continuous observables

The fundamental prescriptions of quantum theory have so far remained incomplete in that there is no satisfactory prescription for the joint probabilities of successive observations of arbitrary sequence of observables. The joint probability formula derived by Wigner is based on the collapse postulate due to Von Neuman and Luders and is applicable only to observables with purely discrete spectra. Earlier attempts to generalize the collapse postulate to observables with continuous spectra have been unsatisfactory as they lead to only finitely additive (and notσ-additive) joint probabilities in general. In this paper a suitable generalisation of the Wigner joint probability formula is proposed, which is completely satisfactory in the sense that it leads toσ-additive joint probabilities for successive observations of arbitrary sequence of observables, consistent with all the other basic prescriptions of quantum theory. This general law for quantum mechanical joint probabilities is arrived at by a reformulation of earlier results on expectation values in successive measurements. The generalized Wigner joint probability formula is also shown to be a consequence of a general collapse postulate, which allows for changes in state due to measurement from normal states to non-normal states also. As an illustration of our results, the probability distribution of the outcomes of a momentum measurement which immediately succeeds a position measurement is computed, and this seems to shed an entirely new light on the uncertainty principle.

Neutron resonance treatment with details in space and energy for pin cells and rod clusters

Two new codes for calculating with space and energy details in the resonance region are presented here: RMET21 and RMET22. The first one is for homogeneous mixtures and pin cells and the second for cluster type fuel elements. The methods used in these codes are based on the solution of the integral form of the Boltzmann transport equation, with first-flight collision probabilities. Resonance profile tabulations are used, constructed from basic evaluated nuclear data files. In the RMET21 program, there is an option to calculate resonance cross-sections with the Breit—Wigner formula, using any set of resonance parameters. This option is useful, e.g. in comparisons with analytical methods and to make sensitivity studies. In the RMET22 code, a new procedure is presented in the resonance region, using collison probabilities, without the approximation of annularization, commonly used to deal with this geometry. In the RMET22 code, the different pins and other regions within a cluster of rods are treated allowing for the details of space and energy. A new method is used to obtain collision probability tables for interpolations and other approximations to speed up the calculations. Comparisons of results obtained by different treatments and experiments are included, showing the accuracy of the methods presented here and their applicability.