Discrete Eigenstates Research Articles

AlGaAs/GaAs quantum wells with high carrier confinement and luminescence efficiencies by organometallic chemical vapor deposition

Single and multiple quantum wells with well widths ranging from 10 to 135 Å were grown by atmospheric pressure organometallic chemical vapor deposition and characterized by photoluminescence (PL) and electrical measurements. Compared with single layers of high-purity GaAs and AlGaAs which have high intensity near band-edge excitonic transitions, the quantum well (QW) structures exhibit very strong luminescence of the discrete QW eigenstates. The intense QW signals indicate that the carrier confinement efficiency of the wells is very high, which is also supported by significant mobility enhancement in the samples. The sharp PL lines suggest high quality and smoothness of the well interfaces.

Stieltjes orbitals for molecular photoexcitation and ionization spectra: N ? V? and N ? V? resonance features in CO and H2CO cross sections

Computational studies are reported of Stieltjes orbitals for Hilbert-space descriptions of discrete and continuum molecular electronic eigenstates. Particular attention is focused on the identification of N V π(π π*) and N Vσ(σ σ*) intravalence contributions to photoexcitation and ionization spectra in CO and H2CO molecules. Three-dimensional graphical representations of appropriate Stielties orbitals serve to identify the σ σ* transitions of Mulliken as photoionization resonances above threshold in these compounds, whereas the corresponding π π* contributions are spectrally localized as familiar discrete excited states. The development illustrates the nature and properties of Stieltjes orbitals, and demonstrates their utility in studies of both discrete and continuum excitation spectra on a common basis.

Nonrelativistic Coulomb Green’s function in parabolic coordinates

The nonrelativistic Coulomb Green’s function G(+)(r1,r2,k) is evaluated by explicit summation over discrete and continuum eigenstates in parabolic coordinates. This completes the derivation of Meixner, who was able to obtain only the r1=0 and r2→∞ limiting forms of the Green’s function. Further progress is made possible by an integral representation for a product of two Whittaker functions given by Buchholz. We obtain the closed form for the Coulomb Green’s function previously derived by Hostler, via an analogous summation in spherical polar coordinates. The Rutherford scattering limit of the Green’s function is also demonstrated, starting with an integral representation in parabolic coordinates.

Density of states in the gap of a disordered material using E.S.R./optical correlations

Abstract Electron-spin-resonance studies of rapidly frozen solutions of alkali metals in anhydrous hexamethylphosphoramide (HMPA) indicate the presence of several, distinct localized electron states. Optical studies of similar species in a variety of fluid solutions reveal a simple correlation between the optical transition energy and the degree of occupation of the alkali-metal ns orbital. From this information we evaluate a distribution function for the various (localized) paramagnetic states present in the low-temperature disordered system. The localized states assume only a few, discrete eigenstates within the host (HMPA) bandgap. We speculate on the possible origin of these states; particular emphasis is placed upon the influence of lone-pair electrons and steric effects in determining the molecular configurations of the host matrix around the alkali atom during the quenching process. We point out the striking similarities between the eigenfunctions/eigenstates of matrix-bound states and the excited ...

Quantum-mechanical treatment of collision-induced dissociation

A close-coupling technique for calculating quantum-mechanical probabilities of collision-induced dissociation (CID) of a diatomic molecule by an atom is presented. The internal Hamiltonian (of the diatomic) is first diagonalized in a discrete, square-integrable basis. The lowest several of the resulting discrete eigenstates approximate the true bound states and the remaining (pseudocontinuum) states represent the true continuum. Next, the stationary collision wavefunction is expanded in the diagonal basis to obtain a discrete set of close-coupled equations, which are integrated numerically by standard procedures. The method is applied to a collinear model in which the diatomic is bound by a Morse potential and the interaction is a repulsive exponential. The total CID probabilities appear to be converged to 1% or 2% in most cases. Vibrational ’’enhancement’’ of CID is observed in this model. A very general problem associated with the use of the exponential interaction in conjunction with a binding potential which supports a continuum is discussed.

Group theoretic approach to the screened Coulomb problem

Perturbation method based on the group theory is developed for the radial solution of the Schrödinger equation with screened Coulomb potential. It may be treated as an alternative to the analytic perturbation theory proposed recently by McEnnan, Kissel, and Pratt ( Phys. Rev. A, 2 (1976), 532) and is based on the expansion of the potential of the following form V(r) = −( a r ) × (1 + λV 1r + λ 2V 2r 2 + …) . Corrections to the point-Coulomb energy levels are given as series in λ and also the screened Coulomb eigenstates are given in the form of expansions in powers of λ. The method is applied also to the continuous spectrum and similar expansions are found. The problem of the normalization of both discrete and continuous spectrum eigenstates is discussed and we find some differences in the case of the scattering states. Origin of this discrepancy is explained.

Magnetic properties of intermediate impurity states in frozen solutions of alkali metals in hexamethylphosphoramide

Electron spin resonance spectra at 77 K show the existence of several, discrete, intermediate impurity states which differ markedly in the percent occupation of the metal outer ns orbital. It is proposed that an amorphous matrix allows impurity states to assume only certain discrete eigenstates within the bandgap.

Stieltjes imaging of photoabsorption and dispersion profiles

Variational and moment-theory techniques for the construction of accurate approximations to photoabsorption and dispersion profiles in atoms and molecules are described and applied to simple atomic systems. It is shown that appropriately defined principal pseudostates, when employed in ab initio variational calculations, furnish the discrete transition frequencies and oscillator strengths that provide so-called principal representations of spectral moments. The Tchebycheff-Stieltjes-Markoff moment theory ensures that the principal frequencies and strengths furnish the information necessary for constructing convergent images of both the discrete and continuum portions of photoabsorption profiles. Evaluation of the appropriate principal value integrals results in convergent Stieltjes images of the associated photodispersion profiles. Detailed applications in the cases of the negative hydrogen ion and atomic hydrogen show that the Stieltjes imaging procedure is rapidly convergent for both the discrete and continuum portions of absorption and dispersion profiles in simple systems. Additional illustrative applications in the cases of atomic helium and lithium show that photoabsorption profiles with simple structures can be accurately imaged even when the pseudostates employed are not necessarily principal ones. The present development extends previously described variational and moment-theory techniques for approximating frequency-dependent dipole polarizabilities in their analytic regions to the real frequency axis within photoabsorption intervals. Moreover, the Stieltjes theory shows that conventional configuration interaction methods with square-integrable basis functions can be employed in the calculation of principal pseudostates and associated photoabsorption and dispersion profiles, thereby avoiding the explicit construction of accurate approximations to the complete set of discrete and continuum eigenstates required in more customary approaches.

An S matrix description of state mixing: Part I: General formalism, with Rho-Omega mixing as an Illustration

The notion of state mixing is formulated in terms of S matrix language alone. The object is to work with quantities which are closely related to experiment, while avoiding the particularities of Hamiltonian theory or field theory. The scheme applies to particles of any spin, it is explicitly unitary, and useful in phenomenology as well as in dynamical S matrix models. Unlike the mass matrix method, it avoids the assumption that the states to be mixed evolve from discrete eigenstates of an unperturbed Hamiltonian. The main ingredient is a perturbation method by which the inverse of the K matrix is related to a corresponding unperturbed matrix. In some cases the latter can be defined directly from physical scattering matrices, without the device of turning off interactions. In this and following papers the method is applied to ϱ - ω, ϕ - ω, and K 0 - K 0 mixing. The results amount to more than a simple rephrasing of earlier treatments. As the discussion of ϱ - ω mixing in this paper illustrates, the physical arguments take quite a different form, and some new aspects of mixing questions are brought out.

Majorana Equations for Composite Systems

A relativistic wave equation for the description of a composite system with nonrelativistic internal motion is deduced from the Bethe-Salpeter equation. It is shown that when the internal motion can be described by algebraic methods, as in the case of the hydrogen atom, the proposed equation for the motion of the system as a whole (motion of the c.m.) is equivalent to a Majorana-type equation, free from the well-known difficulties such as a spacelike solution. The hydrogen-atom example is discussed in some detail. It is shown how its $\mathrm{SO}(4,1)$ spectrum-generating algebra contains the spin part of the generators of the Lorentz group. The solutions found to the Majorana-type equation, which is able to describe both discrete and continuum eigenstates, span a representation of an $\mathrm{SO}(4,2)$ algebra.

On the quasiparticle description of nuclei

TheT-operator of a stableN-nucleon system is expressed by means of a renormalizable equation. This equation is derived without any approximations on the basis ofWeinberg's quasiparticle theory. Its application to a certain model system yields equations which are very similar to the basic equations used byMigdal andLarkin in their phaenomenological theory of nuclear structure. It can be shown, however, that this model system approximates theN-nucleon system only in a restricted energy region which is different from the region of validity postulated in the theory ofMigdal andLarkin. The reason for this contradiction is the fact that the unperturbed model has discrete eigenstates.

Infrateilchen in der Quantenfeldtheorie

AbstractIt is shown that in the presence of zero restmass particles the spectral assumptions of Quantum field theory have to be modified. Particles coupled to zero rest mass particles may not be eigenstates of the mass operator. In such a case the usual asymptotic condition is not valid and the Hilbertspace has an entirely different structure from the Hilbertspace of “normal” quantum field theory. Quantumelectrodynamics provides an example for this situation. The electron is an “infraparticle”, i. e. an “almost” discrete eigenstate of the mass operator.

A note on the relation between scattering phase and bound states

ABSTRACTThe form of Levinson's relation is obtained for the case where the free Hamiltonian contains discrete eigenstates. The relevance of this to the ambiguous solutions of Low equations and dispersion relations is discussed.