The power radiated by an accelerated charge as it moves along a brachistochrone is shown in this paper to be proportional to (1-$v^2$/$c^2$). This is quite unlike the behavior described in the literature, where, for example, the power radiated varies as (1-$v^2$/$c^2$${)}^{\mathrm{-}2}$, in the case of a circular orbit. The calculations here are a sequel to a recent paper by Goldstein and Bender [J. Math. Phys. 27, 507 (1986)], in which the brachistochrone for a charge falling relativistically in a uniform electric field is worked out.

We show that the continuous family of steadily advancing, shape-preserving solutions of the Ivantsov problem for dendritic crystal growth is not necessarily broken down to a discrete set by the introduction into the model of an effect which defines a natural length scale for the problem. Specifically, we give an exact solution of a model which includes a special anisotropic heat-loss term in the liquid phase, which can be used to define length and velocity scales for the problem. The resulting needle-crystal solutions for a given undercooling of the liquid form a continuous family. There is a definite relation between tip curvature and growth velocity for these solutions, but the latter is allowed to take any value above a finite lower limit.

Synchrotron x rays have been used to produce a confined multicharged ion gas near room temperature. Comparison of charge-state-number observations characteristic of ion formation and of ion storage, together with measurements of Ar-to-${\mathrm{Ar}}^{q+}$ electron-transfer rate coefficients, provide information to estimate time constants for relaxation to thermal equilibrium and other stored-ion properties important to further development of the technique.

Staircase and ladder methods are proposed for atomic and molecular total and dissociation energies. In both methods, the energies are generated by employing indirect paths via information obtained from effective potentials. In the staircase method, the energies are determined in steps by successive alternations of electron and proton removals. Within the Hartree-Fock (HF) staircase formulation, the energy for electron removal is taken as the negative of the highest-occupied orbital energy, and the energy for proton removal is obtained as the difference of conventional HF total-energy expectation values. The HF staircase total and dissociation energies are significantly superior to the traditional HF values. In the ladder method, total energies are obtained by summing successive highest-occupied orbital energies for fixed nuclei. Both methods are useful within more advanced many-body theories and are exact within exact Kohn-Sham density-functional theory where the magnitude of the highest-occupied orbital energy equals the experimental ionization energy. Self-interaction-corrected density-functional results are presented. We assert two new Koopmans theorems: (1) the energy $\mathcal{E}$ of the highest-occupied HF orbital would give the experimental ionization energy $I$ if the exact ground-state wave function were free of single excitations out of this orbital, and (2) $\mathcal{E}=-I$ when the exact correlation potential, $v_c\mathrm{}\left(\right[n];\mathrm{r})$, is added to the Fock potential and self-consistency is achieved in both the HF orbitals and in the density $n$.

The diffraction of sodium atoms from a standing wave of near-resonant light is studied as a function of the atomic velocity component along the k vector of the wave. A simple theory is presented which quantitatively predicts the observed diffraction patterns and the decrease of the rms momentum transfer to the atoms with increasing velocity. This decrease results from the dephasing between the two counterpropagating traveling waves due to the first-order Doppler shift. In addition, these measurements have resolved the previous discrepancy between theory and experiment concerning the overall momentum transferred by a standing wave.

Oscillatory convection in binary fluid mixtures can take the form of traveling waves, in contrast to the stationary flow patterns observed in pure fluids. Using both mechanical and thermal excitation methods, we have been able to create propagating pulses of oscillatory, traveling-wave convection. We report a study of such pulses and measurement of the phase and group velocity, the real and imaginary parts of the dispersion, and the reflection coefficient of the pulses at lateral walls. The collision of two counterpropagating pulses is also described.

The effect of periodic modulation of the vertical temperature gradient on the Hopf bifurcation in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures is studied experimentally. The threshold for the onset of convection is found to be reduced by modulation, with the amount of reduction linearly proportional to the amplitude of modulation. The results can be understood on the basis of a phenomenological model.

The dynamical multifractal spectrum of hyperbolic systems is found to be the fundamental equation in a kind of statistical-mechanics formalism for both permanent and transient chaos. It is shown that the free energy may appear in an eigenvalue problem, the solution to which provides a new method for calculating dynamical spectra. Explicit examples are given and the possibility of extending the method for higher-dimensional systems is discussed.

In many-body theory, the single-particle self-energy can be constructed from the two-body vertex, the single-particle Green’s function, and the interaction in a well-known way. Functional differentiation of the self-energy leads back to the two-body vertex. Given a set of two-body vertex diagrams, the process of constructing self-energy diagrams and functionally differentiating them generates a second set of two-body vertex diagrams. If the initial set of diagrams includes the noninteracting two-body Green’s function, the two sets will be equal if and only if the initial set includes all diagrams. We discuss this result from the perspective of the parquet theory.

The quantum-mechanical many-body problem is studied using the extended coupled-cluster method (ECCM) as a convenient parametrization of the Hilbert space. A systematic development of the formalism is given, and the quantum-mechanical problem is cast into the form of a classical Hamiltonian field theory in a complex symplectic phase space. The equations of motion of the basic ECCM amplitudes are derived both from a dynamical action principle using the ECCM average-value functional, and directly from the Schrödinger equation using the ECCM double similarity transformation. Rules are given for the construction of the Hamiltonian and other observables as well as their products. In particular, commutators are shown to be mapped into generalized classical Poisson brackets. The description is interpreted as an exact bosonization of the quantum theory in which the concept of bosonization is carried to the logical extreme, namely, the resulting generalized coherent bosons are identifiable with classical fields. The quantum-mechanical states of the system are points in the ECCM phase space, and their time evolution, or trajectories, are controlled by a classical Hamiltonian. The bosonization has a perturbation-theoretical basis in terms of maximally linked generalized tree diagrams. The increased degree of locality of the basic amplitudes allows applications to topological excitations and cases with spontaneous symmetry breaking.

This article gives a discussion of the application of the extended coupled-cluster method (ECCM) to the excited states of a general quantum many-body system. The direct eigenvalue equations for the excitation amplitudes of both the ket and the bra eigenstates are derived in the biorthogonal basis obtained by a double similarity transformation. The equations correspond to the diagonalization of a matrix involving second-order functional derivatives of the average-value functional for the Hamiltonian with respect to the basic ECCM amplitudes. The same excitation spectrum is obtained by considering small oscillations around the equilibrium. The problem with its associated effective Hamiltonian has the structure of a generalized random-phase approximation. By diagonalizing the effective Hamiltonian we perform a canonical or symplectomorphic coordinate transformation into normal coordinates in the symplectic ECCM phase space. In this coordinate system the exact average-value functional for the Hamiltonian has a structure analogous to that of classical lattice dynamics or the phenomenological Ginzburg-Landau theory. At all stages the method satisfies the property of quantum locality, which in real space shows up as a definite quasilocality. Due to this property the method allows, for example, the treatment of mesonlike excitations in the presence of topological objects or in other symmetry-broken equilibrium states.

Using a previously formulated theory of quantum electrodynamics (QED) based on self-energy, we give a general method for computing the Lamb shift and related Casimir-Polder energies for a quantum system in the vicinity of perfectly conducting boundaries. Our results are exact and easily extendable to a full covariant relativistic form. As a particular example we apply the method to an atom near an infinite conducting plane, and we recover the standard QED results (which are known only in the dipole approximation) in a simple and straightforward manner. This is accomplished in the context of the new theory which is not second quantized and contains no vacuum fluctuations.

General relations concerning wave operators are proposed that exhibit symmetry between the model space and the target space. The basic Bloch and des Cloizeaux effective Hamiltonians operating in the model space are shown to be equivalent to the exact Hamiltonian operating on well-defined vectors belonging to the target space. For the Bloch formalism, a new perturbative development, involving the usual wave operator, is proposed. It has improved convergence properties and makes possible the calculation, at each perturbation order, of an estimated error in the usual Bloch perturbative development. A numerical test on the simple Mathieu equation, performed at orders 2, 3, and 4, gives promising results.

We analyze the barrier problem and the Kronig-Penney model for Dirac particles, paying attention to the Klein paradox and the nature of the potential. The ambiguities associated with the δ-function limit are examined and the physically reasonable way to include δ-function potentials in the Dirac-Kronig-Penney problem is identified.