Energy levels of a spherical quantum dot in a confining potential

Using an exact analytical method, we obtain the ground state and the excited states wave functions as well as the corresponding eigenvalues of a spherical quantum dot in the presence of a confining potential which is a combination of linear, Coulomb and quadratic terms. Next, we investigate the quantum dot energy as the potential coefficients are changed. Our study reveals that considering such a confining potential leads to results which are in good agreement with experimental results.     

An Alternative Model of the Damped Harmonic Oscillator Under the Influence of External Force

In this paper we introduce the modified time-dependent damped harmonic oscillator. An exact solution of the wave function for both Schrödinger picture and coherent state representation are given. The linear and quadratic invariants are also discussed and the corresponding eigenvalues and eigenfunctions are calculated. The Hamiltonian is transformed to SU(1,1) Lie algebra and an application to the generalized coherent state is discussed. It has been shown that when the system is under critical damping case the maximum squeezing is observed in the first quadrature F _x . However, for the overcritical damping case the maximum squeezing occurs in the second quadrature F _y . Also it has been shown that the system for both cases is sensitive to the variation in the coherent state phase.     

Matter as Spectrum of Spacetime Representations

Bound and scattering state Schrödinger functions of nonrelativistic quantum mechanics as representation matrix elements of space and time are embedded into residual representations of spacetime as generalizations of Feynman propagators. The representation invariants arise as singularities of rational representation functions in the complex energy and complex momentum plane. The homogeneous space GL(²)U(2) with rank 2, the orientation manifold of the unitary hypercharge-isospin group, is taken as model of nonlinear spacetime. Its representations are characterized by two continuous invariants whose ratio will be related to gauge field coupling constants as residues of the related representation functions. Invariants of product representations define unitary Poincaré group representations with masses for free particles in tangent Minkowski spacetime.     

An approach to dark energy problem through linear invariants

The time evolution of vacuum energy density is investigated in the coherent states of inflationary universe using a linear invariant approach. The linear invariants we derived are represented in terms of annihilation operators. On account of the fact that the coherent state is an eigenstate of an annihilation operator, the wave function in the coherent state is easily evaluated by solving the eigenvalue equation of the linear invariants. The expectation value of the vacuum energy density is derived using this wave function. Fluctuations of the scalar field and its conjugate momentum are also investigated. Our theory based on the linear invariant shows that the vacuum energy density of the universe in a coherent state is decreased continuously with time due to nonconservative force acting on the coherent oscillations of the scalar field, which is provided by the expansion of the universe. In effect, our analysis reveals that the vacuum energy density decreases in proportion to t_β where β is 3/2 for radiation-dominated era and 2 for matter-dominated era. In the case where the duration term of radiation-dominated era is short enough to be negligible, the estimation of the relic vacuum energy density agrees well with the current observational data.     

Acousto-exciton interaction in a gas of 2D indirect dipolar excitons in the presence of disorder

A theory for the linear and quadratic responses of a 2D gas of indirect dipolar excitons to an external surface acoustic wave perturbation in the presence of a static random potential is considered. The theory is constructed both for high temperatures, definitely greater than the exciton gas condensation temperature, and at zero temperature by taking into account the Bose–Einstein condensation effects. The particle Green functions, the density–density correlation function, and the quadratic response function are calculated by the “cross” diagram technique. The results obtained are used to calculate the absorption of Rayleigh surface waves and the acoustic exciton gas drag by a Rayleigh wave. The damping of Bogoliubov excitations in an exciton condensate due to theirs scattering by a random potential has also been determined.     

Lie algebraic approach of a charged particle in presence of a constant magnetic field via the quadratic invariant

In this paper we consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be isotropic and the magnetic field is applied along the z-axis. The canonical transformation is invoked to remove the interaction term and the system is reduced to a model containing the second harmonic generation. Two classes of the real and complex quadratic invariants (constants of motion) are obtained. We have employed the Lie algebraic technique to find the most general solution for the wave function for both real and complex invariants. Some discussions related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself are also given.     

AN INTEGRAL EXPRESSION OF EXACT SOLUTION FOR HARMONIC OSCILLATOR WITH TIME-VARYING FREQUENCY

Using Feynman's method of disentangling operators, an exact expression of evolution operator of harmonic oscillator with time-varying frequency is obtained. The parameters in this expression are related to the time-varying frequency directly. Some implications are further investigated, including general form of linear and quadratic invariants and evolution of Wigner function.     

Quantum propagator for some classes of three-dimensional three-body systems

In this work we solve exactly a class of three-body propagators for the most general quadratic interactions in the coordinates, for arbitrary masses and couplings. This is done both for the constant as the time-dependent couplings and masses, by using the Feynman path integral formalism. Finally, the energy spectrum and the eigenfunctions are recovered from the propagators.     

An approach to dark energy problem through linear invariants

The time evolution of vacuum energy density is investigated in the coherent states of inflationary universe using a linear invariant approach. The linear invariants we derived are represented in terms of annihilation operators. On account of the fact that the coherent state is an eigenstate of an annihilation operator, the wave function.in the coherent state is easily evaluated by solving the eigenvalue equation of the linear invariants. The expectation value of the vacuum energy density is derived using this wave function.Fluctuations of the scalar field and its conjugate momentum are also investigated. Our theory based on the linear invariant shows that the vacuum energy density of the universe in a coherent state is decreased continuously with time due to nonconservative force acting on the coherent oscillations of the scalar field,which is provided by the expansion of the universe. In effect, our analysis reveals that the vacuum energy density decreases in proportion to t-β where β is 3/2 for radiation-dominated era and 2 for matter-dominated era. In the case where the duration term of radiation-dominated era is short enough to be negligible, the estimation of the relic vacuum energy density agrees well with the current observational data.     

The partition function of degenerate quadratic functional and Ray-Singer invariants

The partition function of degenerate quadratic functional is defined and studied. It is shown that analytic torsion and similar invariants can be interpreted as partition functions of quadratic functionals.     

Spinor fields with zero mass in unbounded isotropic media

The Dirac equation for massless fields in unbounded media has solutions similar to the focus wave mode solutions of Maxwell's equations leading to infinite dynamical invariants. We define the splash wave mode solutions as a weighted superposition of the focus wave modes, and discuss the conditions to be fulfilled by the weight functions to make the dynamical invariants bounded. We leave open the physical interpretation of these solutions.     

Path integral analysis of harmonic oscillators with time-dependent mass

Two cases of forced harmonic oscillators with time dependent mass for which exact propagators can be evaluated are presented. From the exact propagators, normalized solutions of the corresponding Schrödinger equations are arrived at. Time-dependent invariants are also found.     

The partition function of a degenerate functional

The partition function of a degenerate quadratic functional is defined and studied. It is shown that Ray-Singer invariants can be interpreted as partition functions of quadratic functionals. In the case of a degenerate non-quadratic functional the semiclassical approximation to the partition function is considered.     

Multiple scattering in random media. III. Coherent potential propagators and fluctuations

An earlier microscopic approach to the theory of the averaged resolvent operator for an electron interacting with impurities is formulated in terms of coherent propagators. We study the corrections to the coherent potential approximation arising from fluctuations. For uncorrelated positions of the impurities, the linear, restricted, and general two-body additive approximations to the treatments of fluctuations are studied. For general correlations, the linear and restricted two-body additive approximations are studied. For both coherent and bare propagators, corresponding treatments of fluctuations involve the same correlation functions for impurities.Work supported in part by the National Science Foundation under Contract No. NSF DMR 79-23213.     

Construction of exact invariants of time-dependent linear nonholonomic dynamical systems

In this work, we build exact dynamical invariants for time-dependent, linear, nonholonomic Hamiltonian systems in two dimensions. Our aim is to obtain an additional insight into the theoretical understanding of generalized Hamilton canonical equations. In particular, we investigate systems represented by a quadratic Hamiltonian subject to linear nonholonomic constraints. We use a Lie algebraic method on the systems to build the invariants. The role and scope of these invariants is pointed out.     

Wave Function in Classical Statistical Mechanics

The possibility to formulate classical statistical mechanics in terms of the complex wave function and density matrix obeying the evolution equation is discussed. It is shown that the modulus squared of the introduced wave function of the classical particle has the same physical meaning as the modulus squared of the wave function of the quantum particle. The tomographic probabilities are studied for both classical and quantum states. Integrals of motion and their relation to the propagators are discussed.     

The effect of electron state hybridization on the frequency dependence of hopping conductivity in disordered systems

The effect of electron state hybridization on phononless hopping conductivity in a disordered system of point localization centers in the intermediate frequency range corresponding to electron transitions to distant centers is discussed. When the basis of the atom-type localized functions is applied, with the hybridization of the wave functions of the distant centers not being taken into account, the frequency dependence of phononless conductivity is shown to be superlinear and monotonic in a wide frequency range. In this case, the kink in the vicinity of a crossover from a linear to a quadratic frequency dependence of the conductivity turns out to be sharper as distinct from the standard theory.     

Theory of optical spectra of impurity centres in crystals: general consideration of quadratic vibronic coupling

A method is proposed for the calculation of the spectra of the multiphonon transitions caused by the quadratic vibronic coupling with a phonon continuum. In the method, the time evolution of the final state is described by the path integrals. By applying the Stratonovich-Hubbard identity, the quadratic coupling is presented as the fluctuating linear coupling; the latter is calculated by the Lax method. As a result, the problem is reduced to the calculation of the determinants of matrices, the elements of which are given by the pair correlation functions of the contributing configurational coordinates. The method has been verified for the case of two mixed modes. Numerical calculations have also been made for a modified Debye model.     

Weighted-Residual Methods for the Solution of Two-Particle Lippmann–Schwinger Equation Without Partial-Wave Decomposition

Recently there has been a growing interest in computational methods for quantum scattering equations that avoid the traditional decomposition of wave functions and scattering amplitudes into partial waves. The aim of the present work is to show that the weighted-residual approach in combination with local basis functions give rise to convenient computational schemes for the solution of the multi-variable integral equations without the partial wave expansion. The weighted-residual approach provides a unifying framework for various variational and degenerate-kernel methods for integral equations of scattering theory. Using a direct-product basis of localized quadratic interpolation polynomials, Galerkin, collocation and Schwinger variational realizations of the weighted-residual approach have been implemented for a model potential. It is demonstrated that, for a given expansion basis, Schwinger variational method exhibits better convergence with basis size than Galerkin and collocation methods. A novel hybrid-collocation method is implemented with promising results as well.     

Modal and characteristics-based approaches for modeling elastic waves induced by time-dependent boundary conditions

In this paper, we present a characteristics-based approach for solving elastic wave problems with time-dependent traction boundary conditions. A generalized mathematical model for this important class of problems is expressed as a set of first-order, linear, hyperbolic partial differential equations. We analyze the mathematical structure of this first-order linear system, verify its hyperbolicity, derive its characteristic form, and deduce its eigenvalues, eigenvectors, and Riemann invariants. The eigenvalues correspond to the wave speeds, while the Riemann invariants are used to construct a solution by the method of characteristics.