Simulation of the Hard Core by a Velocity Dependence

A comprehensive phase-shift analysis of ${\ensuremath{\pi}}^{+}\ensuremath{-}p$ elastic-scattering data at 310-Mev incident-pion laboratory kinetic energy has been performed. The experimental data utilized include measurements of the differential and total cross sections and of the recoil-proton polarization. The $D$-wave phase shifts were found to be definitely needed in order to attain an adequate fit to the data. A general search for phase-shift solutions was carried out, using $S$-, $P$-, and $D$-wave phase shifts. One solution\char22{}of the Fermi type\char22{}was found that fits the data significantly better than any of the other solutions obtained. The calculated errors in the phase shifts of this set vary from 0.4 to 0.6 deg. Because it was felt that these errors might be deceivingly restrictive, the effects of small nuclear $F$-wave phase shifts on the results of the analysis were investigated and were found to be large: not only are the uncertainties in the original Fermi-type solution increased, but additional sets of phase shifts arise that fit the data well. One of these new solutions is similar to the original Fermi set except that the magnitudes of the phase shifts in this new fit are in general larger than those in the initial solution, and the signs of the $D$-wave phase shifts are reversed. The nuclear phase shifts in the original Fermi solution and their rms errors are (when $F$-wave phase shifts are allowed): ${S}_{3,1}=\ensuremath{-}17.2\ifmmode\pm\else\textpm\fi{}2.6$ deg, ${P}_{3,1}=\ensuremath{-}2.9\ifmmode\pm\else\textpm\fi{}4.0$ deg, ${P}_{3,3}=135.0\ifmmode\pm\else\textpm\fi{}0.6$ deg, ${D}_{3,3}=3.1\ifmmode\pm\else\textpm\fi{}2.6$ deg, ${D}_{3,5}=\ensuremath{-}4.9\ifmmode\pm\else\textpm\fi{}2.1$ deg, ${F}_{3,5}=0.5\ifmmode\pm\else\textpm\fi{}0.6$ deg, ${F}_{3,7}=\ensuremath{-}0.6\ifmmode\pm\else\textpm\fi{}1.4$ deg. Although theory appears to favor this set, further theoretical and experimental evidence is desirable. The values given here for the first five phase shifts approximate the corresponding values obtained when the $F$-wave phase shifts were assumed negligible. However, all except ${P}_{3,3}$ fall outside the limits set by the small original errors. Inelastic-scattering processes were neglected during the phase-shift analysis. Calculations indicate that, if these processes could properly be taken into account, any changes in the quoted values of the phase shifts would probably be well within the corresponding errors given here. Extension of the phase-shift inquiries to include $G$ waves was attempted, but it was observed that the available data and theory do not allow the $G$-wave interaction to be significantly incorporated into the analysis.

The binding energy of nuclear matter at the equilibirum density determined by Brueckner and Gammel is calculated with the Brueckner-Gammel-Thaler nuclear forces. The Brueckner theory is applied. It is found that a solution exists for the coupled equations which determine the off-energy-shell elements of the $\mathcal{K}$ matrix of the Brueckner theory. This exact treatment of the off-energy-shell effects increases drastically the single-particle potential at high momenta. The calculated binding energy per nucleon, 7.5 MeV, is much smaller than the value of 15.2 MeV obtained by Brueckner and Gammel. This result is discussed. Two methods of treating the repulsive core---hard core and "hollow hard core"---are compared.

Effect of short-range repulsion in nuclear matter

Model dependence of ann' PP Bremsstrahlung, and the binding energy of nuclear matter

The accurate computation of nuclear matter properties, such as its binding energy, is still a challenge to nuclear theory, largely because the quark substructure of nucleons creates a strong repulsion at short distances that renders a straightforward perturbative calculation impossible. In this work, the authors study the Brueckner-Bethe-Goldstone expansion of dense Fermi systems in terms of hole-line contributions, of the associated Goldstone diagrams, to the binding energy of nuclear matter. They use various modern nucleon-nucleon potentials of high precision. In all cases the three-hole-line contributions are sufficiently small suggesting that the expansion converges. Yet, the empirical saturation properties of nuclear matter are not reproduced for any potential. This means that very strong nuclear three-body forces are required in order to achieve satisfactory saturation properties of nuclear matter, and that such forces are essential for all considered potentials.

It has been shown by Kapuy that any alternant molecular orbital wave function is an antisymmetrized product of strongly orthogonal geminals, in which the geminals are of mixed spin. Employing the localized alternant orbitals previously introduced by the author, this fact is used to generate a wave function which is capable of interpolating between the single-parameter AMO wave function and a wave function of the Hurley, Lennard-Jones, and Pople type. The validity of the unprojected form of the least flexible wave function of this type is tested in a model calculation on the hexagonal ring of hydrogen atoms. In the test case, this unprojected wave function is found to do significantly better than the unprojected AMO wave function and almost as well as the projected AMO wave function. It is shown that in the case of an infinite system, this wave function must do better than the projected AMO wave function. Features of the more general wave functions of this type are discussed, and it is mentioned that any of these wave functions may be written as an antisymmetrized geminal powers (AGP) wave function in which the geminal is of mixed spin.

Apresentamos um recurso eletrônico destinado a apoiar o ensino e facultar o estudo independente da dualidade onda-partícula na física quântica. Seguindo outros autores, escolhemos o interferômetro de Mach-Zehnder como ilustração conveniente dos conceitos. O material está organizado em "objeto de aprendizagem", cujo elemento central é uma animação interativa e comentada. Textos auxiliares apresentam o formalismo matemático subjacente bem como informações de caráter histórico, epistemológico e experimental. Três níveis de descrição são considerados sucessivamente na animação. O primeiro mostra o feixe na aproximação da óptica geométrica, enquanto o segundo analisa os aspectos puramente ondulatórios. O terceiro nível apresenta a visão corpuscular quântica, introduzindo pacotes de ondas associados a fótons individuais e explicitando a relação entre contagem e função de onda. A inserção de um detector num dos braços do instrumento permite discutir a clássica questão do caminho seguido pelo fóton. A animação apresenta visualizações dos ingredientes que fundamentam as descrições teóricas, tais como frentes de onda, deslocamentos de fase, sobreposições e separações de pacotes em componentes. Levando ao pé da letra a interpretação mais usual da mecânica quântica, exibimos na tela do computador o "colapso do estado" induzido pela observação. Acreditamos que tais visualizações teórico-conceituais complementam proveitosamente a simulação realista do dispositivo experimental desenvolvida por outros autores.

The present chapter treats the spatial properties of the scattering process as described by the wave function. After an introduction, in Section XVII.2 the differential equation for the radial wave function (the Schrodinger equation) is derived. Section XVII.3 presents the solutions of the free radial wave equation, and lists some of their properties. The properties of the exact radial wave functions, in particular their asymptotic forms and their connection to the phase shifts and the S-matrix, are discussed in Section XVII.4. In Section XVII.5 the connection between bound states and poles of the S-matrix on the positive imaginary axis is established. Some material about functions of a complex variable, which is needed in this chapter and in Chapter XVIII, is reviewed in a mathematical appendix.

Entangled-photon pairs ~biphotons! generated by spontaneous optical parametric down-conversion exhibit a number of properties that are analogous to those of ordinary photons generated by incoherent sources. The spatial pump-field distribution and the two-particle wave function in the biphoton case play the respective roles of the source intensity distribution and the second-order coherence function in the incoherent case. The van Cittert‐Zernike theorem, which is applicable for incoherent optical sources emitting independent photons, has a counterpart for biphotons. Likewise, the partial-coherence theory of image formation has an analogous counterpart for biphoton beams transporting spatial information. However, an underlying duality, rather than analogy, accompanies the mathematical similarity between incoherent and biphoton emissions if the comparison is made between the photon count rate in the incoherent case and the biphoton count rate in the entangledphoton case. The smaller the size of an incoherent source, the more separable is the coherence function and the more coherent is the field, and therefore the higher the visibility of ordinary interference fringes. In contrast, the narrower the size of a biphoton pump source, the more separable is the wave function and the less entangled is the field, and therefore the lower the visibility of biphoton interference fringes. This duality is similar to the complementarity between single and two-photon interference exhibited for biphotons.

Andersen's atomic-sphere approximation has been utilized with approximations based upon linear-combination of atomic orbitals (LCAO) theory to obtain approximate energy-band parameters for solids. Simple analytic expressions for the bandwidth and position of the band center have been derived that require only free-atom wave functions evaluated at the Wigner-Seitz atomic-sphere radius. For convenience, the method has been named the atomic surface method (ASM). The following simple analytic expressions for the band parameters have been derived from the ASM: (i) The bandwidth is equal to the product of ${\ensuremath{\Elzxh}}^{2}$/m, the gradient of the electron density at the atomic-sphere radius, and the surface area of the sphere; (ii) the average band energy is shifted from the atomic-term-value energy by an amount given by the product of the bandwidth, electron density at the atomic-sphere radius, and atomic-sphere volume. The theory has been applied without adjustable parameters to the transition metals and f-shell metals with use of tabulated Hartree-Fock wave functions and is in reasonable agreement with full band-structure calculations. The same analysis is applied to atomic core states under compression and is also in reasonable agreement with complete band-structure calculations. The 2s and 2p states of Na and Al have been calculated to the point where they merge with the conduction band as free-electron states. These bandwidths and shifts are also written in terms of the atomic term values by using the asymptotic form of the radial wave function. Finally, the LCAO energy bands of Ni are calculated with use of the ASM parameters. The average d-band energy relative to the conduction-band minimum and the hybridization of the conduction band with the d band are calculated with use of the asymptotic form of the radial wave function, giving approximate energy bands entirely in terms of free-atom parameters.

The theory of low-energy electron-hydrogen atom scattering is reformulated to include, in any truncated coupled-channel calculation, the contribution from continuum states due to exchange effects. The existence of such a contribution has been explicitly demonstrated by Castillejo, Percival, and Seaton, although the standard treatment does not include it. Our formulation of the problem employs the expansion of the exact wave function ${\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}$ in terms of the complete set of hydrogenic states ${{\ensuremath{\phi}}_{\ensuremath{\alpha}}}$ in the form ${\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}=\ensuremath{\Sigma}{\ensuremath{\alpha}}^{}{\ensuremath{\phi}}_{\ensuremath{\alpha}}(1){{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}(2)$. Although this expansion is not termwise symmetric or antisymmetric, the entire sum is made so by use of the integral equation for ${\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}$. The boundary conditions satisfied by the scattering coefficients ${{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}$ are automatically specified in this approach. Furthermore, it is now clear that the entire, exact elastic or inelastic amplitude is contained in the scattering wave function ${{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}$. The elastic wave function ${{\ensuremath{\psi}}_{0}}^{\ifmmode\pm\else\textpm\fi{}}$ (and also the set of open-channel wave functions) is shown to obey both homogeneous (optical potential) and inhomogeneous equations. The equivalence of these equations with each other and with the equation found using Feshbach's projection-operator method is demonstrated. A set of homogeneous equations are found for the case where only certain chosen channels are kept and the rest are truncated. This latter set of equations is identical to those obtained by Hahn, O'Malley, and Spruch as an adjunct to calculations establishing bounds on scattering parameters. Both the exact and the approximate sets of equations for the ${{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}$ differ from the corresponding equations used in all previous coupled-channel calculations. Using our results, the relation between the Born-Oppenheimer amplitude and first-order exchange amplitude of Bell and Moiseiwitsch is discussed. Each is shown to be a different first-order approximation to the exact amplitude, although the Born-Oppenheimer amplitude is seen to be an approximation to the amplitude derived from the inhomogeneous equation. Some simple single-channel calculations of phase shifts and scattering lengths have been carried out. The triplet phase shifts and scattering length derived from the approximate homogeneous equation are identical to those of the standard static exchange approximation, a result expected on the basis of the antisymmetry of the ${\ensuremath{\Psi}}^{\ensuremath{-}}$. However, the singlet phase shifts and scattering length differ from those of the standard static exchange approximation. We find that for ${k}^{2}\ensuremath{\le}0.09$, the singlet phase shifts are larger than those calculated in the $1s\ensuremath{-}2s$ strong-coupling approximation, and that the singlet scattering length is 7.85, where quantities are given in atomic units. For ${k}^{2}>0.09$, the singlet phase shifts are less than those of the $1s\ensuremath{-}2s$ case and approach those of the static exchange calculation. Reasons for this behavior are discussed.

The effect of vacuum polarization upon $p\ensuremath{-}p$ scattering is considered by first solving the problem in the Coulomb plus vacuum polarization potentials (called "electric" potential) without the nuclear potential. The nuclear phase shifts are then defined with respect to the electric wave functions, and the scattering cross section is written in terms of these phase shifts. The connection with other nuclear phase shifts (in the presence of vacuum polarization) which appear in the literature is established.The effective-range expansion for the nuclear $s$-wave phase shift is derived. An analysis of three low-energy $p\ensuremath{-}p$ scattering experiments indicates that the omission of vacuum polarization from the analysis results in a value for the shape-dependent parameter which is 0.02 smaller than the value obtained when vacuum polarization is included in the analysis. A discussion of the accuracy required to usefully delimit this parameter is included.

We present isospin-zero phase-shift analyses of the world $\mathrm{np}$ data near 325 MeV. Inclusion of new ${D}_{t}$ (and $P$) measurements produces a unique $I=0$ phase-parameter solution; however, $\ensuremath{\delta}(^{3}G_{4})$ is almost 5 standard deviations below theoretical predictions. Fixing this phase shift and the other $G$-wave phase shifts at theoretical values results in a set of lower partial-wave phase parameters which interpolate well with phase shifts at adjacent energies. Theoretical consequences of this phase shift set are discussed.NUCLEAR REACTIONS $E=325$ MeV; nucleon-nucleon isospin-0 phase-shift analysis.

We consider a proposal to define the wave function of the Universe as a sum over spacetimes that eventually inflate. In the minisuperspace model, we explicitly show that a simple family of initial conditions, parametrized by a positive real number $a_0$, can be imposed to realise this prescription. The resulting wave function is found to be proportional to the Hartle-Hawking wave function and its dependence on $a_0$ is only through an overall phase factor. Motivated by this observation, we ask whether it is possible to analytically extend $a_0$ to an extended region $\bar{\mathcal{D}}$ in complex $a_0-$plane, while retaining the Hartle-Hawking form of the wave function. We use the condition for convergence of path integral and a recent theorem due to Kontsevich and Segal, further extended by Witten, to explicitly find $\bar{\mathcal{D}}$. Interestingly, a special point on the boundary of $\bar{\mathcal{D}}$ recovers the exact no-boundary wave function. Following that, we show that our prescription leads to a family of quantum states for the perturbations, which give rise to scale-invariant power spectra. If we demand, as an extra ingredient to our prescription, a matching condition at the "no-boundary point" in $\bar{\mathcal{D}}$, we converge on a unique quantum state for the perturbations.

Previously determined experimental drift velocities Vdr and ratios of lateral diffusion coefficient to mobility DT/µ have been refitted directly with a three parameter modified effective range theory (MER T) representation of the S wave phase shift, a one parameter fit to the P wave phase shift and fixed higher partial wave phase shifts. The MERT representation now extends to 1·0 eV, a threefold extension of the energy range of the MERT fit reported by Milloy et al. (1977). The total cross section derived from the phase shifts is also reported, together with the differential cross section at 1·0 eV which is compared with a previous experimental determination.