A simple approach to solve the time independent Schröedinger equation for 1D systems

An algebraic approach based on unitary algebras to calculate Franck–Condon factors (FCF’s) is presented. The method is based on the unitary group approach, which consists in adding a scalar boson to the $$\nu $$-D harmonic oscillator space, taking advantage of the transformation brackets connecting the energy, coordinate and momentum representations. In this scheme the solutions are given in terms of an expansion of harmonic oscillator functions. In this way the overlaps are expressed in terms of simple scalar product of the eigenvectors. As a benchmark to illustrate our approach the case of two 1D-Morse potentials is presented. Discussions concerned with both non-Condon contributions and the harmonic limit are included. FCF’s involving the 1D-Morse and asymmetric double Morse potentials are also considered. As an application, the FCF’s involving the S–S stretching in the $$\hbox {S}_2\hbox {O}$$ molecule are described in terms of two Morse potentials.

Improvement on the Formula of Molecular Rotational Spectra

In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr‐Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)‐model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 ‐(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = ‐Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)‐model of the HO the three hj correspond to the self‐adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)‐Hamiltonian . For an m‐fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)‐model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken‐type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE‐condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.

The noncommutative harmonic oscillator, with noncommutativity not only in position space but also in phase space, in arbitrary dimension is examined. It is shown that the ⋆-genvalue problem, which replaces the Schrödinger problem in this case, can be decomposed into separate harmonic oscillator equations for each dimension. The two-dimensional noncommutative harmonic oscillator (four noncommutative phase-space dimensions) is investigated in greater detail. The requirement of the existence of rotationally symmetric solutions leads to a two parameter harmonic oscillator which is completely solved in this case. The angular momentum operator is derived and its ⋆-genvalue problem is shown to be equivalent to the usual eigenvalue problem of the ⋆-genfunction related wave function. The ⋆-genvalues of the angular momentum are found to depend on the energy difference of the oscillations in the two dimensions. Furthermore two examples of a symmetric noncommutative harmonic oscillators are analyzed. The first is the noncommutative two-dimensional Landau problem with harmonic oscillator potential, which shows degeneracy in the energy levels for certain critical values of the noncommutativity parameters, and the second is the three-dimensional harmonic oscillator with noncommuting coordinates and momenta.

This paper demonstrates the application of matrix methods based on finite differences and on variational (Rayleigh-Ritz-Galerkin) procedures to solution of the radial Schrödinger equation for bound states of the Morse potential. It demonstrates sources of numerical inaccuracy: truncation, termination, tolerance, and quadrature. Cubic splines, harmonic oscillators, floating Gaussians, and sines are used as basis functions.

The solvent shifts of vibrational overtone spectra are predicted by Buckingham's theory of solvent effects on Raman and infrared spectra to be linear with the overtone number. We test this prediction for liquid N2 and dilute N2 in liquid Ar by extending the theory and by implementing the predictions using molecular dynamics simulations. Changes are made to the representations of both the intermolecular and the intramolecular parts of the total Hamiltonian. The representation of the intermolecular part is extended by including two additional terms of a Taylor series expansion in the vibrational coordinate of the intermolecular energy. We find, however, that their inclusion does not contribute significantly to the predicted overtone shifts for the systems studied here. The intramolecular potential is represented by a Morse potential in contrast to Buckingham's harmonic oscillator plus a cubic perturbation term. In our calculation the only perturbation terms of the Morse oscillator Hamiltonian are four expansion terms of the intermolecular potential in terms of the vibrational coordinate. Accordingly, the basis functions used to calculate the perturbed energy levels are Morse eigenfunctions. The simulations show that Buckingham's prediction is rather accurate for liquid N2 and dilute N2 in liquid Ar, at standard state conditions, and for liquid N2 at high-pressure high-temperature conditions. We find that for N2 there is a small negative contribution due to quantum anharmonic corrections to Buckingham's harmonic solvent induced vibrational frequency shift. We also show how equivalent theoretical results are obtained when the intramolecular potential is represented as harmonic plus anharmonic terms up to the fifth power of the vibrational coordinate (i.e. a Dunham potential), instead of a Morse potential.

Current simulations of ultraviolet-visible absorption lineshapes and dynamics of condensed phase systems largely adopt a harmonic description to model vibrations. Often, this involves a model of displaced harmonic oscillators that have the same curvature. Although convenient, for many realistic molecular systems, this approximation no longer suffices. We elucidate nonstandard harmonic and anharmonic effects on linear absorption and dynamics using a stochastic Schrödinger equation approach to account for the environment. First, a harmonic oscillator model with ground and excited potentials that differ in curvature is utilized. Using this model, it is shown that curvature difference gives rise to an additional substructure in the vibronic progression of absorption spectra. This effect is explained and subsequently quantified via a derived expression for the Franck-Condon coefficients. Subsequently, anharmonic features in dissipative systems are studied, using a Morse potential and parameters that correspond to the diatomic molecule H2 for differing displacements and environment interaction. Finally, using a model potential, the population dynamics and absorption spectra for the stiff-stilbene photoswitch are presented and features are explained by a combination of curvature difference and anharmonicity in the form of potential energy barriers on the excited potential.

In some quantum chemical applications, the potential models are linear combination of single exactly solvable potentials. This is the case, for example, of the equivalent of the Stark effect for a charged harmonic oscillator (HO) in a constant electric field or an HO in an external dipole field. The same occurs with some molecular systems where the determination of the vibrational spectra remains essentially as a numerical task; the Morse and Hulthén potentials are convenient models for the potential energy of a diatomic molecule. In this work, the supersymmetry (SUSY) and shape invariance methods used to determine exactly solvable potentials are extended to obtain, in a closed form, the eigenvalues of exactly solvable combined potentials, as well as their isospectral or generalized partner potentials. Essentially, the proposal is based on the existence of the Witten superpotential [W(x)] associated with single exactly solvable potentials to find the Witten superpotential for the combined potential under study. For instance, the assumption of twoW(x) allows to obtain an exactly solvable combined potential which is given by a linear combination between the former potentials plus a term resulting from the product of the two singleW(x). The corresponding generalized partner potentials comes from the involved Riccati equation when the resulting Witten superpotential for the combined potential is taken as a specific solution. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Iterative schemes play central role in solving large scale simulations in science and engineering. Development of such methods over the past few hundreds of years faces inevitable difficulty of manual design. Herein, we report, for the first time, iterative schemes that are automatically evolved by genetic programming (GP) and outperform the well-known iterative methods. To cope with the diversity of the systems of linear equations, the proposed technique is applied on a sparse system in 1D and 2D domains and on a non-sparse asymmetric system. Our proof-of-principle experiments demonstrate GP evolved schemes that converge up to 4 times faster than the conventional Gauss-Seidel scheme. Our work paves the way towards automatic design of efficient iterative solvers for large scale systems of linear equations.

A temperature sequence of EXAFS measurements was carried out on a highly dispersed Pt/SiO2 catalyst to determine the temperature dependence of atomic structure in the very small supported clusters. A one-shell EXAFS model was fit to the first neighbor oscillations in the catalyst data to determine bond lengths and relative thermal motion (disorder) for an average cluster environment. A Debye model of disorder was employed in the EXAFS analysis. The disorder measured for the catalyst was determined to be 1.3 to 2 times the value determined for bulk Pt over all temperatures. Asymmetry of the radial distributions of nearest neighbors in the largely surface coordinated clusters and the increased atomic disorder lead to underestimates of the nearest-neighbor distance R1 at higher temperatures. The apparent bond distance contraction with temperature is manifest in the data as a retardation in the phase of the first neighbor oscillations. It is demonstrated that model EXAFS functions employing asymmetric interatomic pair potentials (such as the Morse potential) exhibit very similar bond distance contractions to the catalyst data while maintaining similar estimates for the disorder. Finally, a two-shell EXAFS model was fit to the catalyst data to separately determine the motion of the surface coordinated atoms in the small metal clusters.

. We consider the effect of an external bias voltage and the spatial\nvariation of the surface potential, on the damping of cantilever vibrations.\nThe electrostatic friction is due to energy losses in the sample created by the\nelectromagnetic field from the oscillating charges induced on the surface of\nthe tip by the bias voltage and spatial variation of the surface potential. A\nsimilar effect arises when the tip is oscillating in the electrostatic field\ncreated by charged defects in a dielectric substrate. The electrostatic\nfriction is compared with the van der Waals friction originating from the\nfluctuating electromagnetic field due to quantum and thermal fluctuation of the\ncurrent density inside the bodies. We show that the electrostatic and van der\nWaals friction can be greatly enhanced if on the surfaces of the sample and the\ntip there are two-dimension (2D) systems, e.g. a 2D-electron system or\nincommensurate layers of adsorbed ions exhibiting acoustic vibrations. We show\nthat the damping of the cantilever vibrations due to the electrostatic friction\nmay be of similar magnitude as the damping observed in recent experiments of\nStipe \\textit{et al} [B.C.Stipe, H.J.Mamin, T.D.Stowe, T.W.Kenny, and D.Rugar,\nPhys.Rev. Lett.% \\textbf{87}, 0982001]. We also show that at short separation\nthe van der Waals friction may be large enough to be measured experimentally.\n

We discuss a possibility of a pinning of quantized matter wave vortices by optical vortices in specially arranged optical dipole traps. The vortex-antivortex optical arrays of rectangular symmetry are shown to transfer angular orbital momentum to support "antiferromagnet-like" matter waves. The separable Hamiltonian for matter waves is proposed which allows to factorize of the <i>3D</i> - wavefunction exactly in a product of <i>1D</i> - harmonic oscillator's ground state and <i>2D</i> - vortex-antivortex wavefunction. The wavefunction's phase gradient field associated with the field of classical velocities via Madelung transformation forms labyrinth-like structure. The macroscopic wavefunction of periodically spaced BEC superfluid vortices appears to be less influenced by decoherence.

Quantum confinement is studied by numerically solving time-dependent (TD) Schrödinger equation (SE). An imaginary-time evolution technique is employed in conjunction with the minimization of an expectation value, to reach the global minimum. Excited states are obtained by imposing the orthogonality constraint with all lower states. Applications are made on three important model quantum systems, namely, harmonic, repulsive and quartic oscillators; enclosed inside an impenetrable box. The resulting diffusion equation is solved using finite-difference method. Both symmetric and asymmetric confinement are considered for attractive potential; for others only symmetrical confinement. Accurate eigenvalue, eigenfunction and position expectation values are obtained, which show excellent agreement with existing literature results. Variation of energies with respect to box length is followed for small, intermediate and large sizes. In essence, a simple accurate and reliable method is proposed for confinement in quantum systems.

The Schrodinger equation is introduced and applied to some elementary problems: particle-in-a box, harmonic oscillator, angular momentum, and the hydrogen atom. The representation of eigenfunctions and eigenvalues is discussed, employing linear operators and matrices. The quantum theory of spin, as well as the Pauli exclusion principle, are described. These enable a theoretical understanding of atomic structure and the periodic table. The two disparate modes of time-dependence of a quantum system–unitary evolution and collapse of the wavefunction–are contrasted.

A complete analytical approach is used to study the problem of two electrons in aharmonic potential. The effective potential of the combined system is described as thesum of the harmonic and anharmonic potentials. For large separation distancesbetween two electrons, it is possible to ignore the anharmonic contribution andhence the combined system behaves as a harmonic oscillator. The frequency of themodified oscillator depends on the separation distance between the two electrons andon the angular quantum momentum. When there is a small separation distancebetween the two electrons, the anharmonic contributions become significant andthe system does not have an exact solution. The anharmonic constant, however,does not have explicit dependence on the frequency of the harmonic potentialresponsible for the confinement. We give a rough estimate of the contribution ofanharmonic potentials. As the angular momentum increases, we observe that thecontribution of the anharmonic constant becomes negligibly small. The presentapproximate analytical solution is critically compared with other, existing solutions.