Harmonic Oscillator Wave Functions Research Articles

Reconstruction of the quantum state via position distribution and its time derivative.

The determination of the harmonic oscillator wave function from position distribution and its time derivative is investigated. Quite simple definitive equations result for the expansion coefficients of the wave function in the Fock basis and yield a unique solution. The effect of a finite spatial resolution of the measurements is taken into account. {copyright} {ital 1996 The American Physical Society.}

An alternative series solution to the isotropic quartic oscillator inN dimensions

The series solution of theN-dimensional isotropic quartic oscillator weighted by an appropriate function which exhibits the correct asymptotic behavior of the wave function is presented. The numerical performance of the solution in Hill's determinant picture is excellent, and yields the energy spectrum of the system to any desired accuracy for the full range of the coupling constant. Furthermore, it converges to the well-known exact solution of the unperturbed harmonic oscillator wave function, when the anharmonic interaction vanishes.

Validity of variational procedures for resonant states

The behavior of variational procedures for bound states is well understood, when using harmonic oscillator wave functions whose frequency ω is the only parameter. The same procedure, when applied to problems that have resonant states, shows a completely different behavior. By analyzing the s-wave problem for a cavity potential of the form b′δ(r′−a), we show that, as a function of ω, it gives rise to plateaus that correspond to the real part of the resonant energies, assuming values of b′ related to very narrow resonances compared with their separations. If the latter condition holds, the procedure seems generalizable to other types of potentials.

Generalized entropy as a measure of quantum uncertainty

A generalized entropy is used in order to advance a different form of expressing the Uncertainty Principle of Quantum mechanics. We consider the generalized entropic formulation for different pairs of incompatible observables. In particular, we study the number-phase entropic uncertainty measure for the case of coherent states within the Pegg-Barnett theory. We also tackle the situation of operators with continuous spectra, where a correlation functional is calculated in terms of generalized joint and marginal entropies, for harmonic oscillator wavefunctions.

Representation of three-dimensional rotations in oscillator basis sets

Using the generating function technique we calculate the matrix elements of the rotation operator between harmonic oscillator wave functions of different oscillator lengths. Closed analytical expressions for the matrix elements for harmonic oscillator wave functions in Cartesian and cylindrical coordinates are provided. They are given in terms of finite sums and are adequate for computer programs.

Effect of an in-plane magnetic field on the electronic states and intraband transitions in quasi-periodic Fibonacci superlattices

Quasi-periodic GaAs-(Ga,Al)As Fibonacci superlattices under in-plane magnetic fields are studied within the effective-mass approximation. Electron-envelope wavefunctions, magnetic subbands and intraband transition strengths are obtained using a parabolic model for the conduction band, and via an expansion in harmonic-oscillator wavefunctions. Calculations are performed for magnetic fields related by integer powers of the golden mean tau =(1+512/)/2. It is shown that, for magnetic field values scaled by tau 2n, the corresponding magnetic levels, electron wavefunctions and transition strengths essentially exhibit a self-similar or anti-self-similar behaviour as appropriate for a GaAs-(Ga,Al)As Fibonacci superlattice. Also, for a given magnetic field, the interband transition strengths have similar properties to those found for the case of periodic superlattices. The intraband absorption coefficients are calculated for magnetic field values scaled by tau 2n, and theoretical absorption spectra, when properly scaled, are shown to be self-similar (for even n) or anti-self-similar (for odd n), a result which could be experimentally verified in n-doped GaAs-(Ga,Al)As Fibonacci superlattices.

Plasmon-phonon-coupled modes and their lineshapes for a doped semiconductor superlattice

We studied the plasma oscillations of an electron gas and a hole gas in a GaAs doped superlattice. The doped superlattice is modelled to be a one-dimensional periodic sequence of electron and hole layers which are embedded alternately in a homogeneous polar dielectric host medium. The density-density correlation function has been calculated for a model structure in order to study the plasma-phonon-coupled modes. The strong charge carrier-impurity scattering which results from random doping impurities in the electron (hole) layers, has been taken into account by choosing appropriate polarizabilities for intrasubband-intersubband transitions and the harmonic oscillator wavefunctions as electron (hole) envelope functions. Our calculation demonstrates that the presence of random doping impurities in electron (hole) layers significantly affects both electron and hole plasmons. The well behaved plasma modes in a GaAs doped superlattice can be observed for restricted values of wavevector. In the 2D limiting case, the plasma frequency in a doped superlattice is approximately proportional to (q2-qc2)12/, while for a modulation doped superlattice (such as GaxIn1-xAs/GaAsySb1-y) it is proportional to q, for small q-values. qc is the critical value of the in-plane wavevector q. Our calculated lineshapes of coupled plasmon-phonon modes can be well separated from each other and could have reasonable half-widths and peak heights which could be observed experimentally for the right choice of values for the parameters of the doped superlattice.

Magnetic moments and charge radii of decuplet baryons in a field-theoretic quark model.

We make nonrelativistic as well as relativistic estimations of the magnetic moments and charge radii of decuplet baryons in a field-theoretic quark model, where translationally invariant decuplet baryon states are described by constituent quark field operators and harmonic oscillator wave functions. The relativistic estimations of the magnetic moments made are, however, O(\ensuremath{\Vert}p\ensuremath{\rightarrow}\ensuremath{\Vert}/m) corrections over the nonrelativistic contribution where the higher order corrections for the ground states treated here, being small, are neglected. The constituent quark field operators here with a particular ansatz satisfy the equal time algebra and are also Lorentz boosted in a definite manner to describe the baryons in motion. The estimations for the magnetic moments and their ratios, with a single harmonic oscillator radius parameter for the octet as well as decuplet baryons, show a reasonable agreement with the most recent experimental measurements for ${\mathrm{\ensuremath{\Delta}}}^{++}$ and ${\mathrm{\ensuremath{\Omega}}}^{\mathrm{\ensuremath{-}}}$, which have constrained different models of hadrons to explain both. We feel that the final results in this regard, expected in the near future in succeeding experiments, can further constrain different theoretical models. However, because of the lack of experimental data, the estimated charge radii in the present model stand as model predictions which may be verified in future experiments.

Exact wave functions and nonadiabatic Berry phases of a time-dependent harmonic oscillator.

Exact Schr\"odinger wave functions of a time-dependent harmonic oscillator are found in analytically closed forms for the eigenstates of the generalized invariant and the instantaneous Hamiltonian. The cyclic initial state (CIS) and corresponding nonadiabatic Berry phase are also found exactly for a $\ensuremath{\tau}$-periodic Hamiltonian. There may exist $N\ensuremath{\tau}$-periodic CISs and corresponding Berry phases, but the cases with unstable classical motions do not have CISs in which cases Berry phases do not exist.

Magneto-optical absorption spectra and self-similarity of GaAs–(Ga,Al)As quasiperiodic Fibonacci superlattices under in-plane magnetic fields

A theoretical study of the effects of in-plane magnetic fields on the interband optical absorption spectra of quasiperiodic GaAs–(Ga,Al)As Fibonacci superlattices is presented within the effective-mass approximation. The electron-envelope wave functions and magnetic subbands are obtained by an expansion in harmonic-oscillator wave functions. The theoretical optical absorption spectra are calculated for magnetic fields related by integer powers of the golden mean τ=(1+√5)/2. It is unambiguously shown that, for magnetic-field values scaled by τ2n, the corresponding optical absorption spectra essentially exhibit a self-similar behavior, with the width of the peaks increasing linearly with the field, in agreement with the experimental results by D. Toet, M. Potemski, Y. Y. Wang, J. C. Maan, L. Tapter, and K. Ploog [Phys. Rev. Lett. 66, 2128 (1991)].

A microscopic quark model of πN → KΣ reactions for heavy-ion collisions

We investigate the process πN → KΣ in a microscopic quark model for heavy-ion collisions. In the model the effective 3 P 0 quark-antiquark creation and annihilation vertex is applied. The relative motion of the meson and baryon in the initial (and final) state is taken as plane waves, while the internal quark wave functions of the hadrons are described by harmonic oscillator wave functions. Predictions of the model are reasonably in agreement with the available experimental data.

Eigenvalue integrodifferential equations for orthogonal polynomials on the real line

The one-dimensional harmonic oscillator wave functions are solutions to a Sturm–Liouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm–Liouville problem on the whole real line. In this paper it is shown how to characterize an arbitrary set of polynomials orthogonal on (−∞,∞) in terms of a system of integrodifferential equations of the Hartree–Fock type. This system replaces and generalizes the linear differential equation associated with a Sturm–Liouville problem. Our results are demonstrated for the special case of continuous Hahn polynomials.

Relationship between the deformed harmonic oscillator and clustering in light nuclei

Harmonic-oscillator wavefunctions have been used to study the link between stable nuclear configurations found in alpha cluster model and Nilsson-Strutinsky calculations. Densities of harmonic-oscillator wavefunctions corresponding to the stable configurations in the Nilsson-Strutinsky calculations for 24 Mg have been calculated, these reveal the same structure found in the cluster-model calculations.

WEAK RADIATIVE DECAYS OF PSEUDOSCALAR CHARMED MESONS

We consider here radiative weak decays of charmed mesons such as [Formula: see text] and [Formula: see text] in a quark model where in general a translationally invariant SU(6) hadronic state is described by constituent quark field operators satisfying equal time algebra and a harmonic oscillator wave function. The model had its earlier success during its application to a variety of hadronic phenomena, and here also, without any free parameters, it calculates the branching ratios as [Formula: see text] and [Formula: see text]. Such calculations are very close to the estimations of Asthana and Kamal, and thus is consistent with the QCD enhancement of the latter branching ratio with respect to the former.

On meson-exchange and Δ-isobar currents in the two-nucleon photoabsorption mechanism

The role of π- and ϱ-meson exchange currents and dynamic Δ-isobar currents with both π- and ϱ-exchange are studied in two-nucleon knockout reactions. The Gottfried approximation is tested for these different absorption mechanisms by making a comparison between a factorized (Gottfried) model and an unfactorized model. For the unfactorized model, analytical expressions are given for the different absorption mechanisms by using harmonic-oscillator wavefunctions. The formalism is applied to the 16O(γ,pn) reaction in the photon-energy range E γ = 60–300 MeV. Over the whole energy range, the ϱ-meson degree of freedom is found not to be negligible. It is demonstrated that the unfactorized treatment leads to a sizeable reduction of the cross section and to a change of the shape of the angular cross section.

The role of nucleon-nucleon correlations in the quark distributions in nuclei

Using the non-relativistic quark model of the nucleons and considering the quark exchange between at most two nucleons, we calculate the quark distributions for the ground state of the doubly-magic nuclei, i.e. 4He, 16O and 40Ca. We take into the account the effect of two-body nucleon correlations which depend on spin, isospin, total angular momentum, tensor operators and the relative and centre-of-mass coordinates of two nucleons. The harmonic-oscillator wavefunctions are used as the uncorrelated states and a local density approximation is performed to convert the infinite nuclear-matter correlation functions to finite ones. The result is compared with results due to Catara and Sambataro (1992).

Exact wavefunction of the time-dependent damped harmonic oscillator with an arbitrary varying mass and with a force quadratic in velocity under the action of an arbitrary time-varying driving force

This article adopts a Gaussian-type propagator to find the exact wavefunction of a very general time-dependent damped harmonic oscillator with an arbitrary varying mass and with a force quadratic in velocity under the action of an arbitrary time-varying driving force. The results obtained not only generalize all the known results in the literatures but can also be applied to many interesting particular cases.

Static properties of octet baryons in a field-theoretic quark model.

We consider here static properties, such as the magnetic moments, $\frac{{g}_{A}}{{g}_{V}}$, and charge radii of baryons in a field-theoretic quark model, where translationally invariant SU(6) baryon states are described by constituent quark field operators and harmonic oscillator wave functions. The constituent quark field operators of hadrons at rest, with a particular ansatz, satisfy the Dirac-type algebra and are also Lorentz boosted in a definite manner to describe the baryons in motion. Such a description of hadrons was quite successful in explaining varieties of hadronic phenomena and presently we also observe a reasonable agreement with experiments, wherever available. We may note that the present investigation does not contain any free parameters as they are prefixed by the experimental data as well as from the earlier applications of the model.

Local representation of the exchange nonlocality in n-16O scattering.

The nonlocal Schr\odinger equation is solved rigorously in a microscopic folding model, incorporating both direct and knock-on exchange potentials, for n${\mathrm{\ensuremath{-}}}^{16}$O scattering at laboratory energies of 20 and 50 MeV. The model uses the complex and density dependent n-n interaction of N. Yamaguchi et al. uses harmonic oscillator wave functions for the bound nucleons, and calculates the scattering wave function for this nonlocal problem using a Bessel-Sturmian expansion method incorporating correct boundary conditions. All spins are neglected. The local phase-equivalent potential is obtained from the scattering matrix elements at a given energy by using the iterative perturbative inversion method. This representation allows comparison between the microscopic model and a phenomenological potential, showing good agreement for the local real part of the potential at 20 MeV. From the ratio of the wave functions for the nonlocal potential and for the potential calculated by inversion, a Perey damping factor (PDF) is obtained which is of similar form to the well-known Perey-Buck prescription for the PDF for a Gaussian nonlocality of the conventional range of 0.85 fm. The significance of these results for distorted wave Born approximation calculations is discussed.

Induced transitions and energy of a damped oscillator.

Damped harmonic-oscillator pseudostationary-state wave functions are used to calculate transitions probabilities for a simple harmonic oscillator (SHO) subjected to damping from time t=0. The mean and variance of the energy operator are obtained in the state which was initially an SHO stationary state. This state is an instantaneous eigenstate of the energy at times when sin(\ensuremath{\omega}t)=0, \ensuremath{\omega} being the reduced frequency.