Integrable Hamiltonians of type $H=\frac{1}{2}{p}_x^2+\frac{1}{2}{p}_y^2+{\mathrm{Cx}}^{\alpha +2\mathrm{}}+x^{\alpha }{y}^2$ are discussed. Using the parameters appearing in the Painlevé analysis we find that the integrable potentials for $\alpha =1\mathrm{}$ (Hénon-Heiles) and $\alpha =-\frac{2}{3}$ (Holt) are in one-to-one correspondence. Simple relations exist between the corresponding coefficients $C$, and also the dimensions of the second invariants are found to be related. Quantum integrability is also discussed.

The newly established topological-effect-on-molecular-orbitals (TEMO) theorem has been subjected to a detailed perturbational analysis; this is done in a novel manner by means of the powerful perturbational-variational Rayleigh-Ritz (PV-RR) procedure. TEMO predicts relationships between the molecular-orbital (MO) energy patterns of topologically related $\mathcal{S}$ and $\mathcal{T}$ molecular isomers. The surprisingly strong effect of pure topological factors on the physics of the molecules is demonstrated by the excellent agreement between TEMO predictions and nonempirical self-consistent-field MO calculations. Application of the PV-RR procedure in conjunction with recently introduced radius-of-convergence considerations enables us to dissect in fine detail the influence of perturbations on the TEMO patterns, and thus, for the first time, to study quantitatively the relationship between the topological and physical factors in molecular calculations.

Charge-exchange collision in the cathode fall region of an abnormal glow discharge is assumed to be the mechanism which limits the energy of the ions and generates the energetic neutrals bombarding the cathode. The model used by W. D. Davis and T. A. Vanderslice [Phys. Rev. 131, 219 (1963)] for the calculation of the ion distribution was not applied to calculate the distribution of neutrals. A transport formulation is proposed that allows the calculation of both the distributions of energetic ions and fast neutrals.

A closed-form expression for $1s-1s$ electron capture has been developed in the strong-potential Born (SPB) approximation. Terms of the order ${\left(\frac{Z_p}{v}\right)}^2$ are ignored in our expression, where $Z_p$ is the charge of the projectile and $v$ is the collision velocity. Our errors of order ${\left(\frac{Z_p}{v}\right)}^2$ are within the accuracy of the SPB approximation itself, which is valid to first order in the projectile-electron interaction $V_p$ (and all orders in the stronger target potential $V_T$). Calculations using our expression are in better agreement with experimental observations of the shape of the Thomas peak than are other calculations.

We have observed electric-field-induced resonances riding on a strong autoionizing resonance in the photoionization of excited krypton. We measured the variation of the spacings and strengths of the resonances as a function of their quantum numbers. Our results indicate that up to 3.5 kV/cm, the Stark interaction and the non-Coulombic short-range interaction leading to the autoionizing structure are independent.

We propose and demonstrate quantitatively that a two-component electron distribution with the energy of the suprathermal tail approximately matched to the excitation energy of the upper lasing state results in greatly enhanced x-ray gain in neon-like ions. Such a configuration may be attainable in laser-driven plasmas.

The observation of a new type of saturation effects in degenerate four-wave mixing in Na vapor is reported. The experimental configuration is neither counterpropagating nor velocity selective, and yet sub-Doppler high-resolution lines are observed at the atomic-transition frequencies, and at the weighted center of a Doppler-broadened distribution. These new features are the result of an interplay between absorption and saturation, and the inhomogeneous broadening is essential for their existence.

A unified treatment of spontaneous and stimulated emission in a free-electron laser is presented. Computer simulation results for the time evolution of the laser field, from incoherent spontaneous emission to coherent stimulated emission and eventual saturation, are reported for a one-dimensional model of the Stanford University free-electron laser.

Optical phase conjugation resulting from the two-photon coherence due to the excitonic molecule resonance in CuCl has been observed and studied as a function of frequency, polarization, and pump intensity. Reconstruction of the polarization of an aberrated probe beam is demonstrated.

We extend the direct approach of Lewis and Leach to quantum theory. For a quantum Hamiltonian $H=\frac{P^2}{2+V(q,t)\mathrm{}}$ we find all quantum invariants which are linear and quadratic in the momentum. The form of the potential for each of these invariants is also determined. For the quadratic case we arrive at the quantum Ermakov system for which exact solutions to the Schrödinger equation have recently been discussed.

Energy eigenvalues accurate to eight to six significant figures are obtained for the exponentialcosine screened Coulomb (ECSC) potential with the use of a numerical method. Oscillator strengths for the $2p$ to $1s$ transition are calculated for some values of the screening parameter. Results are obtained for scattering resonances for the $1s$, $2s$, and $2p$ states. The critical screening parameter for the $1s$ state of the generalized ECSC potential has also been calculated.

We discovered a strong hyperfine dependence of the resonant formation process of $d\mu d$ mesomolecules, while detecting neutrons from muon-catalyzed fusion in pure deuterium gas at 34 K. This new effect enabled us to observe directly transitions between hyperfine states of the $\mu d$ atom for the first time and to determine an accurate experimental value for this transition rate. Our analysis demonstrates the importance of hyperfine effects for the quantitative understanding of the mechanism of resonant $d\mu d$ formation. Moreover, this experiment indicates that the resonant formation process is a powerful tool for a refined spectroscopy of $d\mu d$ bound states. Finally, the detailed knowledge about mesoatomic and mesomolecular processes obtained in this work provides valuable information for the analysis of experiments on the elementary muon-capture process in deuterium.

The diamagnetic and Stark shifts of strongly perturbed Rydberg spectra such as those of the alkaline earths are expressed in terms of multichannel—quantum-defect parameters. The theory is applicable when the shifts produced by the external fields are smaller than the neighboring Rydberg separations. It is applied to evaluate the diamagnetic shifts in Ba, an atom in which Rydberg and doubly excited channels are strongly mixed. In such complex spectra, the diamagnetic effect can serve a useful purpose as a tool to disentangle the interacting series. The radial sizes and, therefore, the diamagnetic shifts are quite different for singly and doubly excited states even when they are at the same energy. In fact, for the field strengths considered ($\simeq 40$ kG), the diamagnetic contribution of the doubly excited configuration is quite negligible. As a result, the observed shift is a direct measure of the amount of singly excited configuration in the state of interest. Exactly similar conclusions apply to Stark shifts. The last part of this paper considers states of very high excitation further up in the spectrum which are no longer in a perturbative regime. Spectra of atoms in a magnetic field have shown equally spaced resonances near the ionization threshold with characteristic spacings. By the consideration of the effects of electric fields which are also often simultaneously present, a criterion is established for the nature of these spacings depending on the relative strengths of the crossed electric and magnetic fields. The spacing seen in any particular experiment is shown to depend on the mass and quantum defect of the atom, the strengths of the external fields, and experimental conditions such as the temperature of the atomic vapor whose absorption spectrum is being studied.

The $2p_{\frac{1}{2},\frac{3}{2}}$ levels in ${\mathrm{As}}_4$ and the $2p_{\frac{3}{2}}$ level in ${\mathrm{Se}}_{5,6}$ have been measured by x-ray photoelectron spectroscopy with the use of 1486.6-eV ($\mathrm{Al}K\alpha$) radiation. The $2p_{\frac{1}{2}}$ binding energies have been calculated for the atomic species, as well as for atomic phosphorus, as the difference between the total energies of the $2p_{\frac{1}{2}}$ hole state and the ground (neutral) state $\Delta$ SCF ($\Delta$ self-consistent field method), using the Dirac-Fock program of Desclaux, to which is added the respective correlation-energy difference between the two states. The calculated values of 140.70 eV for $\mathrm{P}2p_{\frac{1}{2}}$, 1368.89 eV for $\mathrm{As}2p_{\frac{1}{2}}$, and 1484.88 eV for $\mathrm{Se}2p_{\frac{1}{2}}$ include a 0.7-1.0-eV increase due to the effects of correlation, consistent with earlier work done on the K and Zn atoms. The $2p_{\frac{3}{2}}$ atomic binding energies were calculated similarly with the use of available Dirac-Slater $\Delta$ SCF values. When we combine these with the measured molecular binding energy of ${\mathrm{P}}_4$, ${\mathrm{As}}_4$, and ${\mathrm{Se}}_{5,6}$ we obtain corresponding $2p$ chemical shifts (atom-molecule) of 3.5, 3.0, and 2.4 eV, respectively. Comparison is also made between the atomic, molecular, and solid-state binding energies of As and Se, and it is found that the solid-state binding energy is the lowest of the three in both cases, as expected from consideration of extra-atomic relaxation.

Error estimates for the approximate complex eigenvalues of the dilated Schrödinger operators are derived. They are calculated for the resonances of the dc Stark effects in hydrogen.

For a harmonic-oscillator gravitational-wave detector, we show that a quantum nondemolition measurement of the square of the number operator may be made by coupling the detector to an oscillator readout via a quadratic interaction, as in optical four-wave mixing. Explicit evaluation of the effect of a meter readout on the detector demonstrates the possibility of arbitrarily accurate instantaneous measurements for sufficiently large coupling strength.

The multipolar formalism is commonly used as the starting point in chemical physics and quantum optics for discussion of the interaction of radiation with atoms and molecules. The relationship of the multipolar to the minimal-coupling formalism is examined when both the electron and the radiation are second-quantized fields. Both the Lagrangian and Hamiltonian formulations are considered: in the former the transformation between the two is a point transformation on the electron field coordinates, while in the latter it is a canonical transformation. The resulting equations of motion are Maxwell's equations, in terms of the basic and auxiliary fields, for the electromagnetic field and Schrödinger equations for charges in an electromagnetic field with the coupling given through the multipole moments. That the Schrödinger equation is different from that which arises in the minimal-coupling formalism is a natural consequence of the use of new field coordinates. The theory is extended to a system of molecules anticipating the discussion of intermolecular energies in paper III (the second succeeding paper).

The electromagnetic field operators and the electron field operators for the coupled system governed by the multipolar Hamiltonian are obtained within the Heisenberg picture. Their causal behavior and their relationship to the minimal-coupling forms are discussed. The basic fields of the multipolar theory, namely, the displacement vector and the magnetic field, are calculated up to terms quadratic in the multipole moment sources. The terms linear in the transition moments are the quantum counterparts of the classical fields and do not change the photon occupation number. The quadratic terms have no classical analogs: they act in both the photon and electron occupation-number spaces. It is shown that it is necessary to include these second-order terms in the calculation of the Poynting vector for an emitting dipole, thus demonstrating their role in the transport of radiative energy.

The multipolar formalism for a system of molecules developed in paper I (first preceding paper) is applied to a molecular pair to calculate the rate of energy transfer and intermolecular potential energies. In these calculations, one of the molecules is treated as passive when placed in the Maxwell field of the other. The electromagnetic field is calculated with the use of the Heisenberg picture as in paper II (second preceding paper). For resonance transfer between identical molecules, it is sufficient to consider the first-order fields to obtain the Förster rate. For nonidentical molecules, the probability of excitation transfer as a function of time is found with the use of the Heisenberg fields. A noteworthy feature of the calculations of dispersion energy is that with the use of electromagnetic fields, correct to quadratic terms in the transition moments, the complete Casimir-Polder intermolecular potential energy can be obtained by the consideration of one molecule as a test polarizable body in the field of the other.

A large enhancement in the resonant collisional energy-transfer process $\mathrm{Na}\mathrm{}\left(\mathrm{ns}\right)+\mathrm{Na}\mathrm{}\left(\mathrm{ns}\right)\mathrm{}\to \mathrm{Na}\mathrm{}\left[\right(n-1\mathrm{}\left)p\right]+\mathrm{Na}\mathrm{}\left(\mathrm{nl}\right)\mathrm{}$ or $\mathrm{Na}\mathrm{}\left(\mathrm{ns}\right)+\mathrm{Na}\mathrm{}\left(\mathrm{ns}\right)\mathrm{}\to \mathrm{Na}\mathrm{}\left(\mathrm{np}\right)+\mathrm{Na}\mathrm{}\left[\right(n-1\mathrm{}\left)l\right]$ is observed when the initial $\mathrm{ns}$ state lies midway between $\mathrm{}(n-1\mathrm{})p$ and an $\mathrm{nl}$ state or between $\mathrm{np}$ and an $\mathrm{}(n-1\mathrm{})l$ state. This resonance condition, achieved by the electric field tuning of the levels, leads to large cross sections ∼ ${10}^9$ ${\mathrm{\AA }}^2$ and narrow linewidths ∼ 1 GHz. These cross sections and linewidths are comparable to the previously observed resonant process $\mathrm{Na}\mathrm{}\left(\mathrm{ns}\right)+\mathrm{Na}\mathrm{}\left(\mathrm{ns}\right)\mathrm{}\to \mathrm{Na}\mathrm{}\left[\right(n-1\mathrm{}\left)p\right]+\mathrm{Na}\mathrm{}\left(\mathrm{np}\right)\mathrm{}$. We have also observed a similar collisional energy transfer starting from a completely different initial state, i.e., $\mathrm{Na}\mathrm{}\left(15d\right)+\mathrm{Na}\mathrm{}\left(15d\right)\mathrm{}\to \mathrm{Na}\mathrm{}\left(16f\right)+\mathrm{Na}\mathrm{}\left(15p\right)\mathrm{}$. In this case the enhancement in the cross section (∼ ${10}^8$ ${\mathrm{\AA }}^2$) is observed at a relatively low electric field ∼ 17 V/cm.