Formula For Phase Shift Research Articles

Low-energy phaseshift formula for e+or--atom elastic scattering

Under the assumption that an incident e+or- interacts with an atom via an effective one-body potential for r d, the authors have derived an exact and practical phaseshift formula for the analysis of low-energy e+or--atom elastic scattering.

On the calculation of phase shifts produced by complex potentials

A formula for the complex phase shift, pertaining to the one-turning-point scattering problem with a complex potential, is given in a general form based on a kind of arbitrary-order phase-integral approximation. The accuracy of the formula is illustrated by application to a real Coulomb potential supplemented by an imaginary potential term proportional to $\frac{1}{{r}^{2}}$. For that case the arbitrary-order phase-integral expression for the complex phase shift is evaluated in a closed, analytical form, which displays the simplicity and accuracy of the higher-order corrections.

Scattering by singular potentials of the general type at high energies

The problem of high-energy scattering by strongly singular potentials, having no convincing general solution yet, is attacked by a generalization of the phase approach. The asymptotically correct expression obtained is recognized as the standard WKB phase-shift formula.

Electromagnetic waves in two-dimensionally inhomogeneous media*

The investigation of electromagnetic waves in two-dimensionally inhomogeneous media leads to an approximation that decouples Maxwell’s equations. The two equations obtained for the longitudinal field components are free from the inconsistencies of the scalar wave approximation. For cylindrically symmetric media the resulting radial equations are basically equivalent to those already derived by Kurtz and Streifer, but a more refined WKB-type approximation of the solutions, using the “method of comparison equations,” leads to the result that all guided modes are split with respect to scalar propagation constants. Calculations carried out for quadratic media indicate that this splitting does not decrease with frequency in the way predicted by previous treatments, using first-order perturbation. The present theory is equally valid for scattered modes, for which an approximate phase shift formula is provided.

Aspects of exact dynamics for general solutions of the sine-Gordon equation with applications to domain walls

A knowledge of the dynamics of nonlinear systems with multiple excitations is important for calculation of the continuum density of states and for one method of calculation of the classical statistical mechanics of the sine-Gordon Hamiltonian. Hirota's $N$-soliton construction for the sine-Gordon equation is extended by analytic continuation to incorporate usefully breather and continuum solutions as well. The significance of these classical solutions is illustrated in the context of the propagation and excitations of domain walls in a classical uniaxial ferromagnet of infinite extent. From these solutions, formulas for the relative phase shifts in the scattering of a soliton, breather, or continuum solution from any one of these, acting as scatterer, are derived. Following Hirota, it is shown that under certain assumptions about the asymptotic properties of the solution the relative phase shifts of solutions corresponding to many soliton, breather, and continuum states may be found by adding the derived pairwise phase shifts.

Correction for the Uniform WKB Wave Functions and the WKB Phase Shifts

A method of improving the uniform WKB solution for single turning point problems is discussed and the correction formulas for the WKB phase shifts are obtained. The method involves the equation of motion which is in a form similar to that used by Fröman and Fröman to obtain the connection formulas in the WKB approximation. A sequence of solutions to the equation of motion is obtained in a remarkably compact form—and in a novel way—by utilizing the involutionality of certain 2 × 2 matrices appearing in the equations. A numerical test of the result thus obtained is presented in comparison with the corresponding numerical solution result. Unlike the ordinary WKB method, the present method is completely free of the connection formulas of the WKB approximation.

Eikonal expansion

An eikonal expansion of the potential scattering T matrix is evaluated, without approximation, through third order in the inverse momentum. Based on the results, their correspondence with the WKB approximation and a new statement of the unitarity constraint, we propose a sequence of four approximations to the exact impact parameter (Fourier-Bessel) representation of the scattering matrix. The sequence consists of the Glauber approximation and three systematic corrections to the Glauber approximation. The corrections are analytic functions of the impact parameter for Yukawa and Gaussian potentials; they vanish for a Coulomb potential. The sequence of eikonal amplitudes is convergent at high energy and is clearly established for small momentum transfer. Validity for all momentum transfer is conjectured based on systematic cancellation, explicitly verified through third order in the expansion, of momentum transfer dependence in the eikonal impact parameter representation. Such cancellation is shown to occur in the explicit construction of the eikonal expansion of the second Born amplitude for a Yukawa potential. Numerical tests of the sequence of eikonal amplitudes show systematic increase of the angular range of validity by comparison with partial wave results for continuous potentials; the theory is not convergent for discontinuous potentials. The WKB phase shift formula is shown to produce a systematic connection with eikonal expansion results. From this we deduce a generating function for the eikonal phase corrections of arbitrary order and also conjecture a sum of the eikonal expansion valid in the limit of high energy and arbitrary potential strength.

Time-Dependent and Time-Independent Potential Scattering for Asymptotically Coulomb Potentials

We consider the three-dimensional quantum mechanical problem of nonrelativistic potential scattering, where the Hamiltonian is H2 = H1 + V = H0 + Vc + V;H0 is the free particle Hamiltonian, Vc is the Coulomb potential, and V(x) is a real-valued potential function defined for x ∈ R3. We show the existence of the wave operators W±(H2,HD) = s-lim eiH2t e−iHD(t) on L2(R3) for V = V2 + V′, where V2∈L2(R3),V′∈L∞(R3)∩Lp(R3)(p<3),and e−iHD(t) is the family of unitary operators used by Dollard to show the existence of the wave operators W±(H1,HD) appropriate for the pure Coulomb case. If V is spherically symmetric then W±(H2,HD) are shown to be absolutely continuous complete. If in addition V(r) is continuous in (0,∞), then W±(H2,HD) are continuous complete. In both cases S=W+*W− is unitary. The connection between the more physical, time-dependent wave operator approach and the traditional time-independent method is made, and phase shift formulas are obtained for the wave and scattering operators.

Exact solution of the Dirac equation with a central potential

The exact solution of the Dirac equation with a central potential, in the semi-relativistic approximation, is derived and formulae for phase shifts and eigenvalue equations are given.

Uncertainties in tests of long-range charge independence

It is pointed out that determinations of the pion-nucleon coupling constant g based on distant collisions in proton-proton (p-p) and neutron-proton (n-p) scattering employing pseudoscalar (ps) theory have questionable validity because the physical particles are not the idealized undressed particles to which the one-pion exchange (OPE) considerations really apply. Neither the pion-pion (π-π) interaction nor the electromagnetic form factors of nucleons (N) and the π's are explained by the derivation of the OPE interaction. Furthermore the Fermi-Yang pion model suggests directly the entrance of derivatives in the π-N interaction energy even though it had to be modified through the years. Although the literal applicability of the ps theory to the calculation of OPE effects is doubtful the exponential dependence of the equivalent OPE potential (OPEP) on the N-N separation is on the other hand a consequence of the tunneling phenomenon in the transfer of a π from one N to another, such as is present in the theory of nucleon-transfer reactions in low energy nuclear physics. The qualitative support for the mathematical form of the OPE from N-N scattering data is thus not a convincing argument for any one detailed mathematical form of OPE theory. It is pointed out that the presence of the factor ( m π/M) 2) in the OPE phase shift formula is of questionable validity. Estimates are made of the effect of disregarding the variation of the factor with m π. The estimates indicate that the π-N coupling constants derived from p-p and n-p data will agree appreciably better with each other as a result. This modification does not destroy the highly probable dependence of the OPE phase shifts on m π expected from the picture of π-tunneling through the region of negative kinetic energy between the nucleons. The modification is equivalent to the replacement of the ps coupling theory by the pv theory employing the factor ƒ/ m π in the interaction energy with actual values of m π corresponding to π +, π −, and π O . It is suggested that tests of long-range charge independence performed by comparing ( g 2) p- p with ( g 2) n- p do not emphasize sufficiently the rather likely possibility of differences in the coupling of the charged pions to both n and p as compared with that of the neutral pion. Possible uncertainties in the coupling constant determinations caused by the characteristics of experimental data are mentioned and briefly discussed. Different forms of the π-N interaction energy are considered. It is pointed out that there is no compelling reason for considering only local forms in necessarily phenomenological formulations of the theory, because the nucleon, pions, and other mesons involved (ϱ, η, μ, ϕ) are not definitely elementary particles and are coupled to each other. The limited meaning of charge independence of the π-N interaction without the incorporation of a quantitative explanation of the pion mass differences is emphasized. It is pointed out that the employment of the pv coupling theory made in this paper is consistent with relatively recent generalizations of that theory made by S. Weinberg and J. Schwinger in connection with PCAC, soft pion production, π-π interaction, nucleon magnetic moments, and related questions. The relationship of matters discussed to theories of electromagnetic mass differences is briefly considered.

On the phase-shift formula for the scattering operator

On the phase-shift formula for the scattering operator

Fredholm Determinants Applied to Electron-Hydrogen Scattering

The properties of the Fredholm determinant for the electron-hydrogen atom system are discussed. These properties are used to derive formulas for the elastic-scattering phase shifts. $S$- and $P$-wave phase shifts are calculated for low energies and the results are compared with other recent calculations. Exchange effects are fully accounted for in the formalism, and a formula is given for the scattering amplitude when two exit channels are present.

Quantum Effects in Small-Angle Molecular-Beam Scattering

Quantum-mechanical calculations of the differential cross section for the small-angle elastic scattering of heavy particles are carried out to establish more definitely the region of validity of the classical approximation. Four results are discussed: (1) The Massey—Mohr phase-shift formula corresponds to the Kennard small-angle scattering formula in the semiclassical limit. (2) The Schiff approximation for the cross section is exactly the same as the semiclassical approximation at small angles, for any central potential. (3) At very small angles the semiclassical limit for the differential cross section varies as exp (—cθ2), where c is a function of velocity for which explicit expressions are given. (4) The first quantum deviation from the classical limit, which is proportional to ℏ2, can be combined with the preceding result to give a reasonable representation of the differential cross section over the entire range of small angles for which quantum deviations are appreciable. Detailed calculations for some specific systems are made, and it is shown that Wu's misgivings over the classical interpretation of the experimental results of Amdur and co-workers are unjustified.

Empirical Analysis of theH−Photodetachment Cross Section

The zero-range photoelectric cross section of Bethe and Peierls, corrected according to effective range theory, is fitted to the ${\mathrm{H}}^{\ensuremath{-}}$ photodetachment cross section. The value of $2.64 {a}_{0}$ obtained for the effective range agrees with values obtained from the wave-function calculations of Schwartz, Burke and Schey, and others. A more accurate formula for the detachment cross section is obtained by allowing for distortion in the final $p$ state by means of an approximate formula for the phase shift. This formula agrees qualitatively with the close-coupling theory phase-shift results of Burke and Schey, and provides a photodetachment cross section that fits the experimental data slightly better than the results re cently obtained by Geltman with an elaborate variational treatment. There are two adjustable parameters: the binding energy, and one parameter in the phase-shift formula. By a slightly different choice of the parameters than is required to fit the experimental data, close agreement can also be obtained with the Geltman results.

On a Paper of Green and Lanford

The phase shift formulas for the wave and scattering operators in the potential scattering of a single nonrelativistic particle are proved under weaker assumptions on the potential than the one used previously by Green and Lanford.

Dispersion-Theoretical Fitting of Nucleon-Nucleon Scattering

A general approach based on partial-wave dispersion relations is presented for obtaining a formula for phase shifts which depends on only a small number of parameters. This approach does not make any specific assumptions on, or approximations of, the unphysical discontinuity, and involves only approximations of functions that are already known from the general framework of the theory. The method approximately reproduces the $^{1}S_{0}$ state in the 0- to 100-Mev range, with two parameters determined from the low-energy data and a third from a phase shift at a higher energy.

GENERAL FORMULA FOR THE SECOND VIRIAL COEFFICIENT

A simple derivation is given of the quantum mechanical expression for the second virial coefficient in terms of the scattering phase shifts. The derivation does not require the introduction of a quantization volume and is based on the identity R(z)−R0(z) = R0(z)H1R(z), where R0(z) and R(z) are the resolvent operators corresponding to the unperturbed and total Hamiltonians H0 and H0 + H1 respectively. The derivation is valid in particular for a gas of excitons in a crystal for which the shape of the waves describing the relative motion of two excitons is not spherical, and, in general, varies with varying energy. The validity of the phase shift formula is demonstrated explicitly for this case by considering a quantization volume with a boundary the shape of which varies with the energy in such a way that for each energy the boundary is a surface of constant phase. The density of states prescribed by the phase shift formula is shown to result if the enclosed volume is required to be the same for all energies.

Shape-Independent Theory of High-Energy Nucleon-Nucleon Scattering

A shape-independent formula for nucleon-nucleon phase shifts, approximately valid over wide energy ranges is derived, assuming only general properties of the wave function and potential. It is found to approximately reproduce the $S$, $P$, and $D$ proton-proton phase shifts in the ranges 10-150 Mev and 10-310 Mev with two, or at most three, free parameters per state. A generalization, which includes part or all of the outer potential exactly, is derived at the same time.

The Phase Shift Formula in the Improved WKB Method

The Phase Shift Formula in the Improved WKB Method

Rigorous Derivation of the Phase Shift Formula for the Hilbert Space Scattering Operator of a Single Particle

For a single nonrelativistic particle moving in a spherically symmetric potential, the existence of the Hilbert space wave operators and S operator is proved and phase shift formulas for these operators are deduced. The probability, P(Ω), for scattering into the solid angle Ω is obtained from the time dependent theory. The relation between P(Ω) and the R matrix of the standard plane wave formulation of scattering theory is established. For collimated incoming packets, it is shown that P(Ω) can be expressed as an energy average of the differential cross section.