Semiclassical Phase Research Articles

Semiclassical model calculations of NH⋯N H-bonds

Using the ab initio MO-LCAO-SCF-CISD method with a 6-31G** basis we calculated, for 30 different N⋯N distances, the NH potentials of an [H 2NH⋯NH 2] − model complex containing a linear NH⋯N H-bond. Based on these potentials we quantized the NH stretching vibration by the semiclassical Planck-Sommerfield phase integral method. Anomalous vibrational spectroscopic behaviour—such as H-bond energies, NH frequencies, anharmonicity, the relation between frequencies and H-bond energies (Badger-Bauer rule), the correlation between NH distances and the N⋯N distances and the NH/ND isotope ratios of the fundamental NH and ND stretching vibration—could be established theoretically over the whole N⋯N distance region, and the proton-transfer mechanism is discussed from the classical point of view. The results are similar to the OH⋯O properties and demonstrate that spectroscopic anomalies of strong H-bonds are not necessarily based on tunnel effects. The success of these results demonstrates, in addition, the usefulness of calculating quantized levels easily from known potentials using the phase integral, without needing knowledge of methods of solving the Schrödinger equation.

The Quantum and Semiclassical Phase Formalism for Obtaining the 1D Eigenfunctions

Abstract The quantum phase formalism, suggested earlier (Abarenov A.V. and Stolyarov A.V. J. Phys. B, (1990) 23 2419-26), for the solution of the one-dimensional (1D) eigenvalue problem, is applied for eigenfunctions and its overlap integrals. The quantum phase and amplitude functions of the offered method is found to concide with semiclassical ones under standart semiclassical conditions (p' << p). It made possible to obtain the closed form of wavefunctions and energies, which is equivalent to those of h2 order W.K.B. expansion. The method is the most efficient for strongly oscillating eigenfunctions, i.e. in the high-energy limit.

Nonlinear evolution of long-wavelength metric fluctuations in inflationary models.

Stochastic inflation can be viewed as a sequence of two-step processes. In the first step a stochastic impulse from short-distance quantum fluctuations acts on long waves---the interaction. In the second step the long waves evolve semiclassically---the propagation. Both steps must be developed to address whether fluctuations for cosmic structure formation may be non-Gaussian. We describe a formalism for following the nonlinear propagation of long-wavelength metric and scalar-field fluctuations. We perform an expansion in spatial gradients of the Arnowitt-Deser-Misner equations and we retain only terms up to first order. At each point the fields obey evolution equations like those in a homogeneous universe, but now described by a local scale factor ${e}^{\ensuremath{\alpha}}$ and Hubble expansion rate $H$. However, the different points are joined together through the momentum constraint equation. The gradient expansion is appropriate for inflation if the long-wave fields are smoothed over scales below ${e}^{\ensuremath{-}\ensuremath{\alpha}}{H}^{\ensuremath{-}1}$. Our equations are naturally described in the Einstein-Hamilton-Jacobi framework, which governs an ensemble of inhomogeneous universes, and which may be interpreted as a semiclassical approximation to the quantum theory. We find that the Hubble parameter, which is a function of the local values of the scalar field, obeys a separated Hamilton-Jacobi equation that also governs the semiclassical phase of the wave functional. In our approximation, time hypersurface changes leave the equations invariant. However, the stochastic impulses that change the field initial conditions are most simply given on uniform expansion factor hypersurfaces whereas propagation is most easily solved on uniform Hubble hypersurfaces, in terms of $\ensuremath{\alpha}({x}^{j},H)$, the nonlinear analog of $\ensuremath{\zeta}$ of linear perturbation theory; we therefore pay special attention to hypersurface shifting. In particular, we describe the transformation process for the fluctuation probability functional. Exact general solutions are found for the case of a single scalar field interacting through an exponential potential. For example, we show that quantum corrections to long-wavelength evolution of the metric are characteristically small using exact Green's-function solutions of the Wheeler-DeWitt equation for this potential. Approximate analytic solutions to our classical system for slowly evolving multiple scalar fields are also easy to obtain in this formalism, contrasting with previous numerical approaches.

Semiclassical phase space description of ionisation and capture for ions colliding with hydrogen-like targets

Semiclassical calculations were performed for 10 keV amu-1-1 MeV amu-1 collisions of protons with H, He+ and Li2+ as well as for He2+, Li3+, C6+ and Ne10+ colliding with hydrogen. The calculated integrated cross sections for global ionisation and capture agree well with classical trajectory Monte Carlo (CTMC) calculations, other theories for capture and with experiment where available. The semiclassical method for the description of the electron processes encounters difficulties in the description of low-energy capture from highly charged hydrogen-like ions.

Strong-field photoionisation of an initially excited hydrogen atom: Formation of Rydberg wavepacket, its structure and trapping of population at Rydberg levels

Strong-field photoionisation from high discrete levels of the hydrogen atom is considered. The secondary coherent population of Rydberg levels via Raman-type transitions is described. The space and time structure of the wavepackets arising is investigated. Sharp jumps of the semiclassical phase of the corresponding wave function are predicted. These features of the wavefunction are used to explain the reasons for its stability with respect to photoionisation in a strong ionising field. An experimental scheme with almost sudden laser excitation of atoms to Rydberg levels is proposed. This scheme is shown to be applicable for observation of the strong-field interference stabilisation of atoms.

Unified semiclassical dynamics for molecular resonance spectra

A method is presented to depict the intramolecular dynamics of resonantly coupled vibrations, starting from the experimental overtone and combination spectrum. The nonlinear least-squares fit of the spectrum is used to obtain a semiclassical phase space Hamiltonian via the Heisenberg correspondence principle. This integrable Hamiltonian, corresponding to quasiperiodic motion, is used to generate a classical trajectory in phase space for each energy level in a resonance polyad. Polyad phase space profiles are shown to have complete mutual consistency starting from a fit in either the local or normal representation. It is argued that the best way to depict the phase space profile is on a spherical surface called the polyad phase sphere. Represented in this way, the local and normal mode phase spaces are seen to be a single entity, manifestly equivalent by a 90° rotation. The phase space trajectories can be converted into a coordinate space representation. This gives an easily visualized picture of the semiclassical intramolecular dynamics corresponding to each energy level. The polyad phase spheres from the fits of the experimental stretching spectra of H2O, O3 and SO2 are displayed. H2O and O3 are seen to be molecules with a local to normal modes transition, while SO2 is seen to be very near the pure normal modes limit. The experimentally determined phase space dynamics of H2O seen on the phase sphere are compared with the dynamics determined by Lawton and Child from trajectory calculations on the Sorbie–Murrell potential surface. The coordinate space trajectories corresponding to the phase spheres are compared with wave functions from the fit of the spectrum.

Semiclassical phase space evolution of Fermi resonance spectra

The evolution of the semiclassical phase space of a Fermi resonance spectrum is investigated as the strength of the resonance coupling is varied between zero and the strong coupling limit. The phase space evolution gives information beyond that contained in the phase space profile of the experimental spectrum alone. The zero-order phase space is found to be different in important respects from that of the pendulum model of a nonlinear resonance. In the weak coupling regime, the phase space evolution is essentially like that of a dynamical barrier picture. In the strong coupling regime of ‘‘intrinsic resonance,’’ the phase space structure is much different. Topology change appears to take place in a more discontinuous manner than in the weak coupling regime. The phase space evolution shows that some levels are problematic for an adiabatic switching treatment. The origin of some anomalous levels seen both in phase space profiles of experimental spectra and in semiclassical quantization studies is clarified.

Computational study and complex angular momentum analysis of elastic scattering for complex optical potentials

A systematic study of elastic scattering by complex valued optical potentials has been carried out. A Lennard-Jones (12,6) potential with an imaginary r−s term has been used in the calculations. The collision parameters are chosen to model the elastic scattering of K by HBr and Li by HBr. First-order semiclassical single turning point phase shifts are compared with accurate quantum phase shifts and close agreement is found. It is straightforward to calculate semiclassical phase shifts for an optical potential; it is unnecessary to introduce additional approximations (for numerical convenience) as is usually done in the literature. Up to 15 000 terms in the partial wave series are used to calculate elastic angular distributions. It is shown that a rich interference structure and a backward glory can occur in the large angle scattering, provided the transition from absorbency to transparency is sufficiently rapid. Different optical potentials can result in similar angular distributions. A complex angular momentum (Regge pole) analysis of the large angle scattering has been carried out. It is shown that the interference effects have a physical interpretation in terms of surface waves that propagate around the core of the potential and the directly reflected elastic scattering.

Semi-classical approaches to the phase space evolutions in intermediate energy heavy ion collisions

The properties of semi-classical phase space evolution equations — as the Vlasov/Boltzman equations — are discussed in the context of the heavy ion reaction theory at intermediate energies (from 10 to 100 MeV per nucleon). The generalized coherent state set is shown to form a (over) complete basis for the phase space ; then every solution of the Vlasov/Boltzmann equations can be defined as a convolution product of the generalized coherent state basis by an appropriate weight function w. The uniform approximation for w is shown to provide an accurate semi-classical description of fermion systems in their ground state : the examples of fermions in a harmonic well and of cold nuclei are discussed. The solution of the Vlasov equation amounts to follow the time evolution of the coherent states which play the rôle of a moving basis. For the Boltzmann equation, the collision term is taken into account by explicit or implicit variations of the function w. Typical applications are discussed : nuclear response to the giant monopole resonance excitation, fast nucleon emission in heavy-ion reactions.

Quantum theory of multiwave mixing IV Effects of cavities on the spectrum of resonance fluorescence

In the presence of a high-finesse cavity, the resonance fluorescence spectrum can be altered by stimulated emission and multiwave mixing processes. We show that under some conditions it is possible for spontaneous emission to induce conjugate fields similar to semiclassical phase conjugation, requiring the use of the multiwave quantum theory described in earlier papers in this series. The theory is applicable to laser spectroscopy and optical bistability. We present explicit spectra and suggest possible experimental tests. In particular we find a quantum limit for FM spectroscopy that shows that even if the medium under study leaves the FM phase and amplitude relationships intact according to semiclassical theory, spontaneous emission can create an AM component detectable in a square-law detector.

Accurate semiclassical determination of rigid asymmetric rotor eigenvalues

Higher-order semiclassical phase integral methods are applied to the determination of rigid asymmetric rotor eigenlevels from the eigenvalues of Lamé’s equation. The method yields accurate eigenvalues for moderate J, J larger than ten, but is especially valuable for very large J because the accuracy of the method improves with increasing J with practically no increase in the computational cost per eigenlevel.

Semi-classical phase space approach for the charge equilibration fluctuations in heavy ion reactions

The fluctuations of the isobaric distributions in deep inelastic collisions are related to the in-phase excitation of the isovector giant dipole resonances of each nucleus. The characteristics of the zero point motion are obtained by the phase space approach method. The calculations are performed for various systems as a function of the interdistance between the collision partners. Estimations are given for density overlap corresponding to the top of the fusion barrier.

Diatomic partition functions from semiclassical phase integrals

Diatomic rotation-vibration partition functions are evaluated through a technique which utilizes semiclassical phase integrals to estimate the eigenvalues. For the assessment of the contributions from the bound and metastable levels, the method is computationally efficient and virtually exact for any potential at any temperature.

Calculation of higher-order semiclassical scattering phase shifts: The effect of a barrier maximum

Evaluation of high-order semiclassical scattering phase shifts is extended to situations involving a potential barrier. Calculations by using appropriate semiclassical phase shift formulas yield accurate results for energies above and below as well as in the immediate neighborhood of the barrier maximum.

An investigation of the exponential atomic collision model: analytical results

A simple, but versatile, atomic collision model for treating non-adiabatic phenomena associated with curve-crossings and non-crossings is the exponential model of Nikitin. This two-state model leads, within a semiclassical framework, to a collisionS-matrix made up of classical trajectory transition probabilities and semiclassical phases. It can be used to discuss physical processes such as excitation and charge transfer. In applying the model expressions to real situations, phase-integral expressions for theS-matrix elements are needed. Such expressions have been obtained earlier by Crothers through a process of interpretation and abstraction. To shed new light on this interpretation we give a phase-integral derivation of the two-state semiclassical collisionS-matrix within a general exponential model. Only analytical results are presented here; numerical ones will be reported in future publications. Representing theS-matrix as a product of two ‘half-way house’ matrices and assuming only one principal transition zone in each half we may parametrize the elements ofSin a semiclassically consistent way. The parameters are computed in terms of certain Stokes constants and complex Coulomb phases arising from the classical trajectory equations describing the evolution of the diabatic (l. c. a. o.) electronic states. The Stokes constants and complex Coulomb phases associated with the canonical (pure exponential model) one-pole, two-transition point problem are parametrized via the comparison equation method, supplemented by strong-coupling asymptotics. The corresponding quantities for a perturbed canonical form (general exponential model) are suitably abstracted in terms of simple known functions and physically significant phase integrals. In this way the complete semiclassical collisionS-matrix is derived. A short discussion of its applicability is given.

Collisions and umbilic catastrophes

The theory of uniform approximations in semiclassical scattering theory is reviewed in order to emphasize the role of the unfolding of the relevant catastrophe type. Application of the theory to situations involving hyperbolic and elliptic umbilic catastrophes is shown to involve a mapping from four physically determined semiclassical phases {fk } to three control (a, b, c) and one translational parameter d. The paper is devoted to solution of this mapping problem. The final reduction is to a ninth degree polynomial equation in a 3, each root of which gives rise to six symmetry related combinations (b, c). Algebraic and numerical investigations demonstrate the general existence of three physically acceptable (real a, (b + c)) combinations for any set of four phases. Given the four associated jacobians {Jk }, this ambiguity may be definitively removed by following the evolution of (a, b, c) between members of a family of related scattering transitions. Numerical experiments indicate that the correct choice...

Evaluation of higher-order semiclassical phase integrals: Application of Chebyshev series approximation and Taylor series expansion

A new approach for evaluating semiclassical phase integrals of the form ∮ dr f(r)/[E−U(r)]k+1/2 for any smooth potential U(r) is derived and tested. The new methods are based on a Chebyshev series or a Taylor series expansions of the integrand and do not explicitly involve handling weights required by existing forms of this procedure. In addition, they are much more flexible and easier to use and incorporate realistic error estimation.

Higher-order semiclassical quantization of an asymmetric double minimum potential

Although the semiclassical quantization condition of a general (asymmetric) double minimum potential has been known for some time it has only been applied to the simpler symmetric case. In this work, new versions of a method for evaluating higher-order semiclassical phase integrals are applied to semiclassical quantization of an asymmetric double minimum potential. The quantization condition is discussed and energy eigenvalues for a model potential are determined in the first- , third- , and fifth-order phase integral approximations. Agreement of three, five, and seven significant digits, respectively, where the exact quantum mechanical eigenenergies are obtained.

Evaluation of higher-order semiclassical phase integrals close to potential extrema

Recently developed techniques for evaluating higher-order semiclassical phase integrals are extended to allow the computation of phase integrals close to potential extrema, where the earlier procedure is numerically unstable. The new methods are required in the calculation of certain integrals encountered in double minimum and resonance scattering problems. Various distributions and ranges of the integration mesh points are studied and global recommendations for use of various versions of the general integration procedure and the choice of mesh points are given.

Comparison of quantum and semiclassical time delay resonance energies and widths: the effect of nearby inner turning points

Semiclassical three turning point time delay calculations of resonance energies and widths have been carried out for the long lived resonance states that lie well below a barrier maximum in the effective potential energy curve of a Lennard-Jones (12, 6) potential. The calculations are based on an alternative uniform semiclassical phase shift and time delay function that allow for the proximity of the inner pair of turning points, unlike previous semiclassical calculations which have allowed for the coalescence of the outer pair of turning points. For narrow resonance states, it is shown that the new semiclassical time delay calculations are in better agreement with accurate quantum mechanical time delay resonance energies and widths than are the previous semiclassical calculations. In addition, some simple approximate semiclassical expressions for the resonance energy and width of a long lived state have been derived, and their accuracy tested.