The generator coordinate method as an approximation to the exact diagonalization in a given model space

The low-lying states of the nuclei ¹⁶O and ²⁰Ne are calculated with the generator coordinate method. For ¹⁶O two generator coordinates are used, which are related to the quadrupole and to the octupole deformation of the single-particle potential. For ²⁰Ne the generator coordinates are related to the parameters β and D of the Nilsson potential. In both cases six pairs of coordinate values are chosen. The results are compared with those of the complete diagonalization. The agreement is, for both cases, better than 0.5 MeV for most low-lying states.     

Quantum knots

We construct an exactly solvable example of Sturmian bound states which exist in the absence of any confining potential. Their origin is topological—these states are found to live on certain “knotted” contours C(N) of complexified coordinates.     

Redundancy coefficients for vibrations involving atoms with four bonds

Simple expressions for the first-order redundancy coefficients in vibrational problems involving atoms with four bonds are given. The general case of no local symmetry is treated as well as the specialization to O_h, C_3v, and C_s symmetry. The elements of the U matrix resulting from the orthogonalization of the local symmetry coordinates with respect to the redundancy condition are given for two suitable sets of symmetry coordinates. The theoretical results are illustrated with the numerical values for ethylsilane.     

Inter-state coupling and definitions of bright and dark states

Contrary to a standard definition of diabatic states (i.e., those without momentum-dependent coupling), based on the construction from adiabatic ones, we defined diabatic states as bright and dark states of a given experiment. Namely, they are defined as states providing maximum, respectively, zero value of electronic transition dipole moments projected to a given polarization vector. Second, the state from (or to) which the optical transition is performed is not from the space of investigated electronic excited state manifold, but it is chosen by the observer. It is shown, for this case, that the inter-state coupling is a general function of vibrational coordinates. The explicit dependence of the inter-state coupling on vibrational coordinates is particularly important for system with strong Stokes shift. The role of exact definitions of bright and dark states as well as the inter-state coupling is discussed with respect to the coherent structure of electronic population observed in optical spectroscopy.     

Integrals of the motion,green functions,and coherent states of dynamical systems

The connection between the integrals of the motion of a quantum system and its Green function is established. The Green function is shown to be the eigenfunction of the integrals of the motion which describe initial points of the system trajectory in the phase space of average coordinates and moments. The explicit expressions for the Green functions of theN-dimensional system with the Hamiltonian which is the most general quadratic form of coordinates and momenta with time-dependent coefficients is obtained in coordinate, momentum, and coherent states representations. The Green functions of the nonstationary singular oscillator and of the stationary Schrödinger equation are also obtained.     

D-dimensional Smorodinsky-Winternitz potential: Coherent state approach

In this study, we construct the coherent states for a particle in the D-dimensional maximally superintegrable Smorodinsky-Winternitz potential. We, first, map the system into 2D harmonic oscillators, second, construct the coherent states of them by evaluating the transition amplitudes. Third, in the Cartesian and the hyperspherical coordinates, we find the coherent states and the stationary states of the original sytem by reduction.     

The quantum mechanical Toda lattice,II

The periodic Toda lattice consists of N particles which move along a closed line and are coupled with an exponential spring to their immediate neighbors. This system, in contrast to the open Toda lattice, has only bound states. In the method of Kac and Van Moerbeke, the classical periodic Toda chain is transformed to a new of set of canonically conjugate variables, μ and ν, which are closely related to the natural coordinates of an open Toda chain with one particle less. The quantum mechanical eigenfunctions for this reduced system are constructed explicitly, and this allows the quantum mechanical analogs of μ and ν to be defined. The bounds states for the periodic Toda chain are then written as linear combinations of functions resembling the wave functions of the reduced open chain. These functions satisfy a set of remarkably simple recursion formulas, and the coefficients in the expansion can be written essentially as a product of as many factors as pairs of conjugate variables μ and ν. Each factor is given as a solution of a second order difference equation which can be recognized as a quantum analog for the equations of motion of one pair μ and ν. The quantization conditions result from cancelling out the exponential growth in the overall wave function, and are phrased in terms of certain phase angles being submultiples of π according as the representation of the group of cyclic permutations. The calculations are simple for N = 3, and moderately tricky for N = 4 although the results are always fairly obvious.     

Polar optical phonon states and dispersive spectra of wurtzite ZnO nanocrystals embedded in zinc-blende MgO matrix

Based on the macroscopic dielectric continuum model and Loudon’s uniaxial crystal model, the polar optical phonon modes of a quasi-0-dimensional (Q0D) wurtzite spherical nanocrystal embedded in zinc-blende dielectric matrix are derived and studied. It is found that there are two types of polar phonon modes, i.e. interface optical (IO) phonon modes and the quasi-confined (QC) phonon modes coexisting in Q0D wurtzite ZnO nanocrystal embedded in zinc-blende MgO matrix. Via solving Laplace equations under spheroidal and spherical coordinates, the unified and analytical phonon states and dispersive equations of IO and QC modes are derived. Numerical calculations on a wurtzite/zinc-blende ZnO/MgO nanocrystal are performed. The frequency ranges of the IO and QC phonon modes of the ZnO/MgO nanocrystals are analyzed and discussed. It is found that the IO modes only exist in one frequency range, while QC modes may appear in three frequency ranges. The dispersive frequencies of IO and QC modes are the discrete functions of orbital quantum numbers l and azimuthal quantum numbers m. Moreover, a pair of given l and m corresponds to one IO mode, but to more than one branches of QC. The analytical phonon states and dispersive equations obtained here are quite useful for further investigating Raman spectra of phonons and other relative properties of wurtzite/zinc-blende Q0D nanocrystal structures.     

Time-dependent harmonic oscillator in a magnetic field and an electric field on the non-commutative plane

In the non-commutative space, wave functions and geometric phases are derived for the time-dependent harmonic oscillator in external time-dependent magnetic and electric field. Explicit forms of the coherent states are also given, which are not the minimum uncertainty states for the coordinates and momenta.     

Overlaps of deformed and non-deformed harmonic oscillator basis states

A systematic approach for expanding non-deformed harmonic oscillator basis states in terms of deformed ones, and vice versa, is presented. The objective is to provide analytical results for calculating these overlaps (transformation brackets) between deformed and non-deformed basis states in spherical, cylindrical, and Cartesian coordinates. These overlaps can be used for reducing the complexity of different research problems that employ three-dimensional harmonic oscillator basis states, for example as used in coherent state theory and the nuclear shell-model, especially within the context of ab initio symmetry-adapted no-core shell model.     

Duality between coordinates and wave functions on noncommutative space

The relation between coordinates and the solutions of the stationary Schrödinger equation in the noncommutative algebra of functions on R^2N is discussed. We derive this relation for a certain class of wave functions for which the quantum prepotentials depend linearly on the coordinates similarly to the commutative case. Also, the differential equation satisfied by the prepotentials is given.     

A study of even-parity states of the nuclei 19F, 21Ne and 23Na with the generator coordinate method and with mixing of projected Hartree-Fock determinants in comparison with complete diagonalization

The energies and electromagnetic properties of the even-parity states of the nuclei ¹⁹F, ²¹Ne and ²³Na are calculated with the generator coordinate method and with mixing of projected HartreeFock determinants, using the Kuo and the Preedom-Wildenthal two-body interactions. The two parameters β and ? of Nilsson's potential are chosen as generator coordinates and the subspace is enlarged with a few different configurations, i.e. one for A = 19 and three for A = 21 and 23 nuclei. Special care is taken in choosing the appropriate Hartree-Fock solution corresponding to possible occupied single-particle states. For both methods six basic functions are enough to obtain a good approximation to complete diagonalization within the same model space. The collective features of these nuclei are also investigated.     

Microscopic description of collective motion in pf shell nuclei

Rotational and vibrational excitations in pf shell nuclei are studied by means of the generator coordinate method. The generator coordinates are the pairing energies and the quadrupole moments of constrained Hartree-Bogoliubov states, projected onto good angular momenta and particle numbers. The Kuo interaction and the one modified by McGrory are used. The vibrational character of the yrast energies appears to be produced by mixing prolate and oblate wave functions. Pairing correlations are essential for this mixing. In contrast to the yrast states the excitation energies of the higher states depend strongly on the interaction used. They show good agreement with experiment, particularly in the case of ⁴⁸Ti with the Kuo force. The calculated B(E2) values exhibit a rotational band structure in general, even if the energies look more vibrational. The force dependence of the excitation energies can qualitatively be understood by inspection of the intrinsic energy surface.     

Resonant tunneling of a few-body cluster through repulsive barriers

A model for quantum tunnelling of a cluster comprised of A identical particles, interacting via oscillator-type potential, through short-range repulsive barrier potentials is introduced for the first time in symmetrized-coordinate representation and numerically studied in the s-wave approximation. A constructive method for symmetrizing or antisymmetrizing the (A ? 1)-dimensional harmonic oscillator basis functions in the new symmetrized coordinates with respect to permutations of coordinates of A identical particles is described. The effect of quantum transparency, manifesting itself in nonmonotonic resonance-type dependence of the transmission coefficient upon the energy of the particles, their number A = 2, 3, 4 and the type of their symmetry, is analyzed. It is shown that the total transmission coefficient demonstrates the resonance behavior due to the existence of barrier quasi-stationary states, embedded in the continuum.     

TDHF equations of motion for algebraic hamiltonians in a coherent state representation

Time-dependent Hartee-Fock (TDHF) equations are derived for nuclear systems with internal dynamical group U(r). The coordinates which appear in the TDHF equations are the coordinates which parameterize the U(r) coherent states. The TDHF orbits for the hamiltonian $\text{H}$ are identical with equations of motion for a classical system described by the hamiltonian function 〈$\text{H}$〉 obtained directly from the operator $\text{H}$. This quantum-classical correspondence facilitates interpretation of TDHF orbits. The phenomena of coexistence and critical elongation are discusses, as is the relation between the critical points of the function 〈$\text{H}$〉 and the spectral properties fo the operator $\text{H}$.     

Radial coherent states—From the harmonic oscillator to the hydrogen atom

We construct spectrum generating algebras of SO(2, 1) ～ SU(1, 1) in arbitrary dimension for the isotropic harmonic oscillator and the Sturm-Coulomb problem in radial coordinates. Using these algebras, we construct the associated radial Barut-Girardello coherent states for the isotropic harmonic oscillator (in arbitrary dimension). We map these states into the Sturm-Coulomb radial coherent states and show that they evolve in a fictitious time parameter without dispersing.