Incoherent X-ray scattering by atoms in semiclassical approximation

The Fourier transform is used for the evaluation of the semiclassical wavefunction in momentum representation. As a consequence of this approach, the incoherent scattering function is obtained.     

Higher order JWKB expressions for the energy levels and the wavefunction at the origin

We derive an exact quantization condition for the energy levels of a particle in a radial potential assumed finite at the origin. This is used to derive corrections to the semiclassical JWKB quantization condition. We further relate the normalization integral of the wavefunction to the energy derivative of wavefunction at origin and use this expression to derive the corrections to the semiclassical JWKB expressions for the wavefunction at origin. An application to upsilon leptonic decay width is also given.     

Divergence-free WKB theory

We present a divergence-free WKB theory, which is a new semiclassical theory modified by nonperturbative quantum corrections. Conventionally, the WKB theory is constructed upon a trajectory that obeys the bare classical dynamics expressed by a quadratic equation in momentum space. Contrary to this, the divergence-free WKB theory is based on a higher-order algebraic equation in momentum space, which represents a dressed classical dynamics. More precisely, this higher-order algebraic equation is obtained by including quantum corrections to the quadratic equation, which is the bare classical limit. An additional solution of the higher-order algebraic equation enables us to construct a uniformly converging perturbative expansion of the wavefunction. Namely, our theory removes the notorious divergence of wavefunction at a turning point from the WKB theory. Moreover, our theory is able to produce wavefunctions and eigenenergies more accurate than those given by the traditional WKB method. In addition, the divergence-free WKB theory that is based on the cubic equation allows us to construct a uniformly valid wavefunction for the nonlinear Schrödinger equation (NLSE). A recent short letter [T. Hyouguchi, S. Adachi, M. Ueda, Phys. Rev. Lett. 88 (2002) 170404] is the opening of the divergence-free WKB theory. This paper presents full formalism of this theory and its several applications concerning wavefunction and eigenenergy to show that our theory is a natural extension of the traditional WKB theory that incorporates nonperturbative quantum corrections.     

A Uniform Semiclassical Calculation of the Rings in Photodetachment Microscope

We describe a uniform semiclassical theory for the spatial distribution of the photodetached electron of H^- in the presence of a static electric field. In this theory we propagate the photodetached electron wavefunction using a mixed position and momentum coordinate method to large distance where a uniform approximation to the electron flux distribution is calculated. Details are given for the propagation of the mixed coordinate wavefunction with cylindric symmetry and the transformation between configuration space wavefunction and the mixed position and momentum wavefunction.     

Reflection and interference structure in diatomic Franck-Condon distributions

The Franck-Condon distributions for diatomic radiative transitions from a single vibrational level of a given electronic state to all possible levels (bound and free) of a second electronic state exhibit either “reflection” or “interference” structure. In reflection structure there is a one-to-one mapping of peaks in the initial state probability distribution into peaks in the spectrum. No such simple relationship is known for interference structure, originally termed “internal diffraction” by Condon [Phys. Rev.32, 858–872 (1928)]. A semiclassical treatment of the quantum mechanical overlap integrals shows that the condition for reflection structure is a monotonic difference potential in the range of internuclear distance sampled by the initial wavefunction, whereas interference structure occurs when a “polytonic” difference potential is sampled. The qualitative validity of the semiclassical treatment is illustrated through quantum calculations which show that the Franck-Condon integrals accumulate near the points of stationary phase.     

Semiclassical treatment of quadrupole shape effects in the elastic scattering of heavy ions below the Coulomb barrier

A perturbative semiclassical approximation for the elastics-matrix is used to derive simple and accurate formulae describing the effects of a nuclear quadrupole deformation on the elastic scattering of aligned nuclei. Expressions are derived for the second rank tensor components of the analysing power, the ratios of which turn out to be simple trigonometric functions of the scattering angle in agreement with experimental observations. The approximations are discussed in some detail and higher order corrections are derived. In an appendix we derive a semiclassical first-order perturbation formula describing the effect of a non-central complex perturbation in the presence of a non-perturbatively treated central term. Our formula is in disagreement with some earlier published formulae which fail to treat the real part of the perturbation correctly.     

Semiclassical scattering amplitude in the complex λ-plane

The stationary phase evaluation of the semiclassical scattering amplitude is replaced by the saddle-point method. In a simple model this allows the interpretation of forward-angle oscillations in terms of a pole of the parametrised S-matrix.     

p + p → d + π+ p-wave production

The production of p-wave pions in p + p → d + π⁺ is calculated by introducing into the initial wavefunction NΔ configurations consistent with elastic scattering. Reasonable agreement with experiment is obtained without further adjustment. The Galilean term has only a small effect.     

A semiclassical approach to inelastic scattering from solid surfaces and to the Debye-Waller factor

A semiclassical formulation of inelastic atom-surface scattering is presented. This formulation is a mixture of classical S-matrix theory and a classical path model. A Debye-Waller factor enters this theory very naturally as the probability of elastic reflection in the presence of inelastic channels. Because of its importance the Debye-Waller factor is discussed in some detail. Finally, assuming a simplified model of the gas-surface system, the whole scattering problem is solved analytically.     

The role of the pion cloud in the interpretation of the valence light-cone wavefunction of the nucleon

The pion cloud renormalises the light-cone wavefunction of the nucleon which is measured in hard, exclusive photon-nucleon reactions. We discuss the leading twist contributions to high-energy exclusive reactions taking into account both the pion cloud and perturbative QCD physics. The nucleon’s electromagnetic form-factor at high Q² is proportional to the bare nucleon probability Z and the cross-sections for hard (real at large angle or deeply virtual) Compton scattering are proportional to Z². Our present knowledge of the pion-nucleon system is consistent with Z = 0.7 ± 0.2. If we apply just perturbative QCD to extract a light-cone wavefunction directly from these hard exclusive cross-sections, then the light-cone wavefunction that we extract measures the three valence quarks partially screened by the pion cloud of the nucleon. We discuss how this pion cloud renormalisation effect might be understood at the quark level in terms of the (in-)stability of the perturbative Dirac vacuum in low energy QCD.     

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ? = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.     

Microscopic derivation of a complex heavy ion potential

The energy dependence of a complex heavy ion optical potential is derived in Brueckner Hartree Fock using a local density approximation and a selfconsistent single particle spectrum. The two ions are described by an antisymmetrized cluster model wavefunction. Both real and imaginary part are found to increase with energy. Results are given for the elastic scattering of ⁴⁰Ca+¹⁶O.     

WKB approximation in complex time

The WKB approximation to the one particle Schrödinger equation in time is used to obtain the wavefunction at a given point as a sum of semiclassical terms, each corresponding to a different classical trajectory (real or complex) but ending up at the same point. A method to find out reflection coefficient for processes involving one and two turning points is developed and it is shown that the semiclassical complex analysis reproduces exactly the reflection coefficient that is obtained through the exact solution of the problem. The connection between pair production and reflection amplitude is also shown. The pair production amplitude in a time dependent gravitational background is calculated and it is shown that the vacuum considered in complex trajectory WKB analysis refers to adiabatic vacuum.     

The deuteron dilation effect in n-d break-up

From the Faddeev equations we show that the quasi-free scattering peak shape, in n-d break-up, is determined by an effective deuteron wavefunction. This differs from the normal deuteron wave function ψ(r) by a distortion factor, that produces a smooth cutoff for small r. This justifies the phenomenological cutoff procedure. The same effective wave function will occur in final-state interaction theory.     

开放量子台球体系散射的赝路径半经典近似

本文讨论正方形量子台球的输运性质,考虑电子以费米能量穿过台球区域,在台球出口和入口处对入射和出射波函数采用基尔霍夫散射.采用微扰论的Dyson方程得到半经典格林函数,并把赝路径半经典近似作微扰展开得到体系的传输矩阵元.比较了传输矩阵元的傅立叶变换谱的峰位置与腔内自由电子经典轨道长度,发现在精度允许范围内它们符合的很好.     

A note on multidimensional WKB wavefunctions: Local and global semiclassical approximations

The semiclassical limit of the wavefunction in several dimensions depends strongly on the structure of the underlying family of classical paths. For the general case of multi-valued trajectory fields one has to distinguish between local and global WKB approximations.     

Semiclassical model for accidental predissociation in diatomic molecules

After a short review of possible quantum treatments, the uniform semiclassical theory of Child (J. Mol. Spectrosc.53, 280 (1974)) is applied to the problem of accidental predissociation in a diatomic molecule where a bound state dissociates into a continuum via a coupling with a near-degenerate second bound state. A complete connection diagram and an analytical expression for the lineshape is provided. This expression is used in the diabatic limit to derive the widths and shifts of the resonances. Checks are made for two test cases for which a simplified quantum model leads to analytical formulas. The familiar case in which each resonance shares the initial width of the perturbing state according to its weight in the configuration interaction wavefunction is verified. In the opposite coupling scheme, the “resonance narrowing” phenomenon predicted by the quantum model is also obtained.