Frequency shifts in emission and absorption by resonant systems ot two-level atoms

The frequency shifts in emission and absorption arising from resonant many-body interactions in a system of two-level atoms are discussed from several points of view: (1) in the language of superradiance, Dicke states, quantum electrodynamics and perturbation theory; (2) in the classical-path treatment of gaseous emission, with emphasis on the impact approximation; (3) by means of diagrams related to the temperature-Green's function formalism; (4) in the semiclassical model using the macroscopic Bloch vector; (5) through ordinary classical electromagnetic theory in a linear medium.     

Semiclassical approach to orbital magnetism of interacting diffusive quantum systems

We study interaction effects on the orbital magnetism of diffusive mesoscopic quantum systems. By combining many-body perturbation theory with semiclassical techniques, we show that the interaction contribution to the ensemble-averaged quantum thermodynamic potential can be reduced to an essentially classical operator. We compute the magnetic response of disordered rings and dots for diffusive classical dynamics. Our semiclassical approach reproduces the results of previous diagrammatic quantum calculations.     

Light scattering,origin invariant multipole moments and molecular property tensors: the case of electric-field-gradient-induced birefringence

A comparison of semiclassical and quantum versions of molecular light scattering theory at finite temperatures is presented. A general formulation of the semiclassical radiation model is developed to the point where its relationship to the corresponding QED formalism can be established: the classical scattered electric field is proportional to the same R-matrix element as that obtained from QED for the photon scattering amplitude. The result is valid for non-resonant scattering at T = 0. The semiclassical theory conventionally also inherits aspects of a classical molecular model, principally origin-dependent molecular multipole moments. Origin independent multipoles, and corresponding response functions can be defined if the theory is cast in terms of centre-of-mass and translation invariant internal coordinates. Such a choice of coordinates brings molecular light scattering theory into line with the theory of the molecular Schrödinger equation. This is illustrated for the case of a diatomic molecule. A specific application of these results of current interest is electric-field-gradient induced birefringence (EFGB) for which there are four competing theories in the literature. In this paper we examine the treatment of finite temperature effects in two semiclassical accounts of EFGB in polar molecules and identify a likely source of the discrepancy between them revealed in a recent ab initio computational study.     

Wronskian analysis of resonance tunnelling reactions

A Wronskian formalism is developed for resonance tunnelling reactions. It is shown how the S matrix can be written in terms of Wronskians between solutions of the Schrödinger equation representing incoming and outgoing waves. The method is an adaptation of the Jost function approach to elastic two-body scattering. Formal expressions are derived for the imaginary parts of the energy eigenvalues that arise from the application of complex boundary conditions used in a previous semiclassical analysis. When the imaginary parts are small, so that the resonance is sharp, the S matrix can be written in a Breit-Wigner resonance form. The theory unifies and extends the semiclassical analysis of resonance tunnelling reactions given previously.     

Greensche Funktionen in der Theorie der Kernmaterie

The aim of this article is the representation of the theory of nuclear matter with the technique of many-body Green functions. We have treated the theory of Green functions thoroughly in the first chapters taking support of the functional method, which offers a convenient approach to the construction of the theory. After treatment of the properties of the exact two-point function and of the effective single-particle potential, preconditions and properties of the approximate solutions are discussed. Results achieved todate are treated in chapter 6. Brueckner theory and allied methods are not further discussed, since these are sufficiently known. The theory of linear response, being important for the semi-phenomenological treatment of real atomic nuclei, is presented in appendix A.     

Pair correlation function in a dense plasma and pycnonuclear reactions in stars

The short-range behavior of the pair correlation function in a dense onecomponent plasma (jellium) is investigated. As an intermediate step, the short-range behavior of the classical pair correlation function is obtained. Actually, although the temperature and the density are assumed to be such that the thermodynamic properties are almost classical, quantum mechanics (tunnel effect) always dominates the pair correlation function at short distances. The quantum pair correlation function is calculated by treating the many-body quantum effects by a perturbation theory, and by using a semiclassical approximation based on path integrals. The results are applied to the computation of the nuclear reaction rate in dense stellar matter (pycnonuclear reactions).Laboratoire associé au Centre National de la Recherche Scientifique.     

Tunneling paths in multi-dimensional semiclassical dynamics

In light of the fundamental importance and renewed interest of the tunnel phenomena, we review the recent development of semiclassical tunneling theory, particularly from the view point of “tunneling path”, beginning from a simple one-dimensional formula to semiclassical theories making use of the analytic continuation, in time, coordinates, or momentum, which are the stationary solutions of semiclassical approximations to the Feynman path integrals. We also pay special attention to the instanton path and introduce various conventional and/or intuitive ideas to generate tunneling paths, to which one-dimensional tunneling theory is applied. Then, we review the recent progress in generalized classical mechanics based on the Hamilton–Jacobi equation, in which both the ordinary Newtonian solutions and the instanton paths are regarded as just special cases. Those new complex-valued solutions are generated along real-valued paths in configuration space. Such non-Newtonian mechanics is introduced in terms of a quantity called “parity of motion”. As many-body effects in tunneling, illustrative numerical examples are presented mainly in the context of the Hamilton chaos and chemical reaction dynamics, showing how the multidimensional tunneling is affected by the system parameters such as mass combination and anisotropy of potential functions.     

Aspects of Green's function methods versus self-consistent field theory

The basic methods of solving fully symmetric, nonlinear theories are stated. These are discussed in terms of Green's function methods and self-consistent field theory methods. The equivalence of many-body theory based on Green's functions with quantum field theory, on which the self-consistent field theory is based, is reviewed. A number of similarities, differences, and cautions involved with these methods are determined. In particular, since very often both methods are based upon use of the adiabatic theorem, which is typicallynot applicable to the models under consideration, a deviation in the self-consistent theory is discussed that avoids this problem. A similar idea is used for solution of models with the functional integral method. Ferromagnetic models are used at various places in illustrating some of the ideas. By contrasting these methods further insight may be gained into solving nonlinear, physical theories.     

Some Approximations in the Theorie of Moments and Transitions for Even-Odd Nuclei

In the framework of Green's functions, the theory of moments and transitions for even-odd-nuclei is developed. It is shown, how these quantities are connected with the effective particle-hole force and the generalized linear response function for the neighbour even-even nuclei. The response function describes the change of the single-particle propagator due to an external perturbation. For superfluid systems the Nambu [1] – Engelsberg [2] – Belyaev [3] technique has been used. The connection between earlier treatments is established, especially with the semiphenomenological approach of Migdal [4] and co-workers. Some approximations for the relevant effective particle-hole force emerging from the many-body theory are discussed in detail. Furthermore the generalization of the RPA-approximation for the auto-correlation function of density fluctuations in the case of superfluidity is given.     

Quantenfeldtheorie wechselwirkender Vielteilchensysteme zur semiklassischen Beschreibung von kollektiver Bewegung mit großen Amplituden

By using path integral methods a collective quantum field theory of interacting many-body systems is developed, the classical limit of which is given by the time-dependent mean-field approximation. In this way the mean-field approximation is embedded into the full quantum mechanics and the quantum corrections to the “classical” mean-field approximation can be systematically evaluated. By including the dominant quantum corrections to the mean-field approximation a semiclassical theory of large amplitude collective motions in many-body-systems, which show a highly nonlinear dynamic and are not accessible to perturbation theoretical methods, is derived. The semiclassical theory is developed explicitly for bound states and decay processes like nuclear fission. In the case of bound states this leads to the quantization of the time-dependent Hartree-Fock-Theory, which is demonstrated for a uniform nuclear rotation.     

Multireference second-order Brillouin–Wigner perturbation theory§

The multireference, state-specific, second-order, Brillouin–Wigner perturbation theory (MR-BWPT2) is presented. A posteriori corrections are made which, in the case of a single reference function, recover the well-known formula of second-order many-body perturbation theory, i.e. Møller–Plesset ‘MP2’ theory, and in the multireference case suppress terms which scale non-linearly with the number of electrons in the system. Prototype calculations are reported for the (H₂)₄ model in which the degree of quasi-degeneracy can be varied by changing a single geometric parameter. The calculated total energies obtained by a second-order Brillouin–Wigner treatment are compared with those supported by CAS-MP2 (complete active space self-consistent field followed by second-order Møller–Plesset-like treatment of dynamic correlation effects), by MR-BWCC (multireference Brillouin–Wigner coupled cluster expansion), and by full configuration interaction. MR-BWPT2 provides a theoretical apparatus comparable to the widely used MP2 theory, but which can be applied to quantum chemical problems requiring a multireference formalism.     

Many-body correlations in the problems of electrodynamics of fluids

This paper presents a review of the theory of the many-body correlation phenomena in fluids, in which the collective character of the fluctuations is caused by the long-range interaction of electrodynamic origin. The procedure of the statistical mechanical averaging of the microscopic electrodynamic equations is developed for a classical system of the interacting polarizable molecules with a subsequent account of the molecular correlations. As a result an effective expansion for the refractive index is obtained. The dependence of the refractive index on the thermodynamic parameters near the critical point is investigated using the scaling-law asymptotics for the many-body correlation functions. A molecular theory of the multiple light scattering is suggested. A method of evaluation of the many-body any-order scattering intensity in the critical region is described. Using a resummation procedure, similar to that developed for the light propagation problem, the correlation function expansion is obtained for a classical system of charged and neutral particles, correlations due to the short-range forces are taken into account. The expansion gives series in terms of the effective “dressed” electrostatic potential containing no long-range Coulomb divergencies, nor short-range ones.     

Stability of Semiclassical Gravity Solutions with Respect to Quantum Metric Fluctuations

We discuss the stability of semiclassical gravity solutions with respect to small quantum corrections by considering the quantum fluctuations of the metric perturbations around the semiclassical solution. We call the attention to the role played by the symmetrized 2-point quantum correlation function for the metric perturbations, which can be naturally decomposed into two separate contributions: intrinsic and induced fluctuations. We show that traditional criteria on the stability of semiclassical gravity are incomplete because these criteria based on the linearized semiclassical Einstein equation can only provide information on the expectation value and the intrinsic fluctuations of the metric perturbations. By contrast, the framework of stochastic semiclassical gravity provides a more complete and accurate criterion because it contains information on the induced fluctuations as well. The Einstein–Langevin equation therein contains a stochastic source characterized by the noise kernel (the symmetrized 2-point quantum correlation function of the stress tensor operator) and yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. These points are illustrated with the example of Minkowski space-time as a solution to the semiclassical Einstein equation, which is found to be stable under both intrinsic and induced fluctuations.     

Beyond the Semiclassical Description of Black Hole Evaporation

In the semiclassical treatment, i.e. in a classical black hole geometry, Hawking quanta emerge from trans-Planckian configurations because of scale invariance. There is indeed no scale to stop the blueshift effect encountered in the backward propagation toward the event horizon. On the contrary, when taking into account the gravitational interactions neglected in the semiclassical treatment, a new UV scale could be dynamically engendered and could stop the focusing. To show that this is the case, we use the large-N limit, where N is the number of matter fields. In this limit, the semiclassical treatment is the leading contribution. Nonlinear gravitational effects appear in the next orders and in the first of these, the effects are governed by the two-point correlation function of the energy–momentum tensor evaluated in the vacuum. In this case they can also be obtained by considering light propagation in a stochastic ensemble of metrics whose mean fluctuating properties are determined by this two-point function.     

Thermal helium clusters at 3.2 Kelvin in classical and semiclassical simulations

The thermodynamic stability of⁴He_4–13 at 3.2 K is investigated with the classical Monte Carlo method, with the semiclassical path-integral Monte Carlo (PIMC) method, and with the semiclassical all-order many-body method. In the all-order many-body simulation the dipole-dipole approximation including short-range correction is used. The resulting stability plots are discussed and related to recent TOF experiments by Stephens and King. It is found that with classical Monte Carlo of course the characteristics of the measured mass spectrum cannot be resolved. With PIMC, switching on more and more quantum mechanics. by raising the number of virtual time steps results in more structure in the stability plot, but this did not lead to sufficient agreement with the TOF experiment. Only the all-order many-body method resolved the characteristic structures of the measured mass spectrum, including magic numbers. The result shows the influence of quantum statistics and quantum mechanics on the stability of small neutral helium clusters.     

A generalized Boltzmann equation for nuclear dynamics

We derive a generalized Boltzmann equation from an extended time-dependent mean-field theory, which is self-consistent and incorporates two-body collisions due to the residual interaction. Obtained from a systematic reduction of a more general, but complicated, many-body theory, this kinetic equation retains many of the quantum effects of the many-body system. Due to proper treatment of quantum causality, its collision integral contains terms which are associated with the principal-value parts of propagators in a quantum-mechanically correct memory kernel. Thus, it properly generalizes the Uehling-Uhlenbeck equation and provides a quantum kinetic theory for nuclear dynamics in both low- and intermediate-energy regions. The new features in this Boltzmann equation are investigated in a nuclear-matter model with a simple effective interaction. We solve for the small-amplitude solutions corresponding to the response of the system to an arbitrary weak external field. The results are contrasted with the collisionless limit and Uehling-Uhlenbeck limit and conclusions are drawn about the dynamical effects of the two-body collisions on the quasiparticles.     

A unified semiclassical description of elastic scattering by a central field

The diffraction-integral formulation of the semiclassical limit of the quantal wavefunction, as proposed in an earlier paper, is applied to the treatment of elastic scattering by a general central-field potential. Angular-momentum quantisation is shown to be a natural consequence of the theory and partial-wave series representations of the s-matrix and asymptotic wavefunction are derived. In addition, the method and results establish a connection between refractive, diffractive, dynamical and uniform-approximation theories of semiclassical scattering.     

Part II. Algebraic tails in three-dimensional quantum plasmas

We review various exact results concerning the presence of algebraic tails in three-dimensional quantum plasmas. First, we present a solvable model of two quantum charges immersed in a classical plasma. The effective potential between the quantum charges is shown to decay as 1/r ⁶ at large distances r. Then, we mention semiclassical expansions of the particle correlations for charged systems with Maxwell-Boltzmann statistics and short-ranged regularization of the Coulomb potential. The quantum corrections to the classical quantities, from orderh ⁴ on, also decay as 1/r ⁶. We also give the result of an analysis of the charge correlation for the one-component plasma in the framework of the usual many-body perturbation theory; some Feynman graphs beyond the random phase approximation display algebraic tails. Finally, we sketch a diagrammatic study of the correlations for the full many-body problem with quantum statistics and pure 1/r interactions. The particle correlations are found to decay as 1/r ⁶, while the charge correlation decays faster, as 1/r ¹⁰. The coefficients of these tails can be exactly computed in the low-density limit. The absence of exponential screening arises from the quantum fluctuations of partially screened dipolar interactions.     

Solution of the Vlasov equation for collective modes in nuclei

A semiclassical theory of giant resonances based on the Vlasov equation is developed. The linearized Vlasov equation is solved for the bound motion of particles in a central potential with an external time-dependent multipole field. The solution obeys an RPA-type integral equation. If the time-dependent part of the self-consistent field is neglected, the solution of the Vlasov equation has a simple analytical form. The strength function for each multipole can be expressed in terms of the natural frequencies of classical orbits and of radial integrals over the classical motion. The method is illustrated by studying the isoscalar monopole, quadrupole and octupole response in medium-heavy nuclei without residual interaction. There are remarkable similarities between the solutions of the semiclassical problem and the corresponding quantum problem. For a central potential with Saxon-Woods shape there is an interesting shift and concentration of strength in the quadrupole and octupole response functions.