Harmonic Oscillator Eigenfunctions Research Articles

Про біквадратичний ангармонічний осцилятор - підхід у рамках розкладу по осциляторному базису. I. Дослідження та розрахунок енергій основного і збуджених станів

For the quantum quartic anharmonic oscillator with the Hamiltonian H=0.5(p2+x2)+λx4 which is one of the classic traditional quantum-mechanical and quantum-field-theory models, its main physical characteristics and properties are thoroughly studied and calculated based on the system's wave function expansion in a complete set of the harmonic oscillator eigenfunctions, i.e., in the basis of eigenfunctions {φ(0)n} of the unperturbed Hamiltonian H=0.5(p2+x2). Very good convergence of the calculated energy levels of the anharmonic oscillator is demonstrated with respect to the number of basis functions included in the expansion, across a wide range of variation of the parameter λ. Thus, we have computed the energies of the ground and the first six excited states of the system for an exceptionally wide range of the oscillator coupling constant λ. In general, the proposed method provides a very good and accurate way to calculate all system characteristics.

Three-dimensional non-approximate Coulomb interaction between two trapped quantum particles

The two trapped quantum particles interacting problem is generalized to three dimensions, and the exact Coulomb potential is used. The system is solved by expanding the wavefunction in terms of the isotropic harmonic oscillator eigenfunctions and Hydrogen atom eigenfunctions independently, showing that each one results in a prime approximation for different domains of the normalized coupling constant of the relative interactions, suggesting that the combination of the basis is enough to build a well-suited base for the non-approximate problem. The results are compared to previous works that use a model of approximate finite-rage soft-core interaction model of the problem to give insights into the many-body states of strongly correlated ultracold bosons and fermions. We conclude that the proposed three-dimensional approach facilitates the distinction between bosons and fermions while the solutions given by the expansions better define the behavior of the particles for repulsive potentials. In addition, we discuss the substantial differences between our work and the previous approximate model.

The propagation of Hermite–Gauss wave packets in optics and quantum mechanics

AbstractThe two‐dimensional paraxial equation of optics and the two‐dimensional time‐dependent Schrödinger equation, derived as approximations of the three‐dimensional Helmholtz equation and the three‐dimensional time‐independent Schrödinger equation, respectively, are identical. Here the free propagation in space and time of Hermite–Gauss wave packets (optics) or harmonic oscillator eigenfunctions (quantum mechanics) is examined in detail. The Gouy phase is shown to be a dynamic phase, appearing as the integral of the adiabatic eigenfrequency or eigenenergy. The wave packets propagate adiabatically in that at each space or time point they are solutions of the instantaneous harmonic problem. In both cases, it is shown that the form of the wave function is unchanged along the loci of the normals to wave fronts. This invariance along such trajectories is connected to the propagation of the invariant amplitude of the corresponding free wave number (optics) or momentum (quantum mechanics) wave packets. It is shown that the van Vleck classical density of trajectories function appears in the wave function amplitude over the complete trajectory. A transformation to the co‐moving frame along a trajectory gives a constant wave function multiplied by a simple energy or frequency phase factor. The Gouy phase becomes the proper time in this frame.Key PointsThis paper builds a bridge between quantum mechanics (QM) and classical optics in that the identity of the paraxial equation of optics and the Schrödinger equation of QM is shown, the Bohmian trajectories of QM are defined in optics, the Gouy phase of optics is defined in QM and given a new interpretation and The space and momentum wave functions are equivalent along a trajectory.

Quantum Anharmonic Oscillators: A Truncated Matrix Approach

This study aims at implementing a truncated matrix approach based on harmonic oscillator eigenfunctions to calculate energy eigenvalues of anharmonic oscillators containing quadratic, quartic, sextic, octic, and decic anharmonicities. The accuracy of the matrix method is also tested. Using this method, the wave functions of the anharmonic oscillators were written as a linear combination of some finite number of harmonic oscillator basis states. Results showed that calculation with 100 basis states generated accurate energies of oscillators with relatively small coupling constants, with computation time less than 1 minute. Including more basis states could result in more correct digits. For instance, using 300 harmonic oscillator basis states in a simple Mathematica code in about 8 minutes, highly accurate energies of the oscillators were obtained for relatively small coupling constants, with up to 15 correct digits. Reasonable accuracy was also found for much larger coupling constants with at least three correct digits for some low lying energies of the oscillators reported in this study. Some of our results contained more correct digits than other results reported in the literature.

Test of an Optimised Exact-Diagonalization Approach for Trapped Fermionic Mixtures

We probe the convergence of the configuration interaction (CI) approach for harmonically trapped one-dimensional Fermi–Fermi mixtures. Existing scientific papers that apply the CI approach to study such systems typically use harmonic oscillator eigenfunctions with fixed frequency parameter values as bases. We show that CI’s optimisation is the key to reducing the computational cost of the numerical study of fermionic mixtures. The effectiveness of the optimised CI approach is analysed in terms of its ability to capture the spatial fermion–fermion correlations.

3D TDHFB theory with Gogny force and fission dynamicsof <sup>240</sup>Pu

Nuclear fission is one of the most complex processes in nuclear physics, involving a strong interplay between nuclear structure and dynamics, as well as various many-body correlations and fluctuation effects. Systematically theoretical and experimental research on the fission problems is of great significance to clarify the fission mechanism, reveal the complexity of low-energy nuclear dynamics, and test the nuclear model theory based on the nuclear many-body theory. However, despite the huge theoretical effort pursued by several generations of physicists, many crucial facets of this process are still not understood at a fundamental level, warranting further exploration of its dynamics. This is more so when viewed from the microscopic theoretical point of view. In the subject of study on nuclear fission, scission dynamics represents one of the most challenging problems in many-body theory as it plays a crucial role for determining the main properties of fission fragments and the emitted light particles. In the scission and post-scission processes, the fragments can experience a rapid change in shape and thus trigger the non-adiabatic effects coming from couplings between collective and intrinsic degrees of freedom. Therefore, from microscopic point of view, a reasonable description of the entire fission process could be achieved with an adiabatic model describing the slow evolution across the barrier, followed by a non-adiabatic treatment of the scission and post-scission dynamics. In this paper, with the aid of the nuclear mean-field model and Gogny effective nucleon-nucleon interaction, a three-dimensional Cartesian mesh code of time-dependent Hartree-Fock-Bogoliubov theory (Gogny-TDHFB Code for Fission) has been developed for solving the static and/or dynamic equations with isolated or periodic boundary conditions and no further symmetry assumptions. In contrast with the Skyrme HFB, the particle-hole channel and the particle-particle channel are treated on an equal footing in the HFB calculations with the Gogny interaction (Gogny HFB). The code consists of two parts: One is the constrained HFB for calculating the adiabatic fission potential energy surface in the multi-dimensional collective phase space, the other is TDHFB for studying the dynamical process of nuclear fission. In order to describe the extremely large deformation of nuclei in the direction of fission, the basis functions in the spatial grid points of the Lagrange mesh are introduced with respect to the direction of the z -axis, and the deformation harmonic oscillator eigenfunctions are used as the basis functions in the ( x , y ) directions. In the calculation of multi-dimensional potential energy surfaces, the constraints are imposed on expectation values of the multi-pole moments Q lm and the augmented Lagrangian method (ALM) is exploited for multi-constrained calculations. A parallel interface using the message passing interface (MPI) library has been implemented for large-scale calculations on massively parallel computers. Under the help of Gogny-TDHFB Code for Fission and by mainly focusing on the scission process, various aspects including the nuclear fission processes, the multi-dimensional potential energy surface, the configuration of scission point, more importantly, the non-adiabatic dynamics in the course of deformation from saddle to scission and also in post-scission are microscopically investigated in order to make clarification on the fission mechanisms, the nature of scission and some other basic problems. As an example, the spontaneous fission process of 240Pu has been studied. The results on the non-adiabatic dynamics in the course of deformation from saddle to scission and also in post-scission show that the pairing correlation plays crucial roles in the scission dynamics of nuclear fission.

Predicting fluorescence quantum yields for molecules in solution: A critical assessment of the harmonic approximation and the choice of the lineshape function.

For the rational design of new fluorophores, reliable predictions of fluorescence quantum yields from first principles would be of great help. However, efficient computational approaches for predicting transition rates usually assume that the vibrational structure is harmonic. While the harmonic approximation has been used successfully to predict vibrationally resolved spectra and radiative rates, its reliability for non-radiative rates is much more questionable. Since non-adiabatic transitions convert large amounts of electronic energy into vibrational energy, the highly excited final vibrational states deviate greatly from harmonic oscillator eigenfunctions. We employ a time-dependent formalism to compute radiative and non-radiative rates for transitions and study the dependence on model parameters. For several coumarin dyes, we compare different adiabatic and vertical harmonic models (AS, ASF, AH, VG, VGF, and VH), in order to dissect the importance of displacements, frequency changes, and Duschinsky rotations. In addition, we analyze the effect of different broadening functions (Gaussian, Lorentzian, or Voigt). Moreover, to assess the qualitative influence of anharmonicity on the internal conversion rate, we develop a simplified anharmonic model. We address the reliability of these models considering the potential errors introduced by the harmonic approximation and the phenomenological width of the broadening function.

Explicit form for the kernel operator matrix elements in eigenfunction basis of harmonic oscillator

In this paper, the matrix elements explicit expressions for the kernel operator in the harmonic oscillator eigenfunctions basis are obtained. The matrix elements are expressed in terms of the two complex variables new polynomials that were constructed in this paper. In the particular case, new polynomials degenerate into Laguerre polynomials. The diagonal elements of the kernel operator matrix are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements contain frequency oscillations responsible for dissipations in the quantum systems.Using the explicit representation of the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems.

Optimized configuration interaction approach for trapped multiparticle systems interacting via contact forces

For one-dimensional systems with delta-contact interactions, the convergence of the exact-diagonalization method is tested with a basis of harmonic oscillator eigenfunctions with frequency Ω optimized through the minimization of the eigenenergy of the desired level. It is shown that within the framework of this approach the well-converged results can be achieved at much smaller dimensions of the Hamiltonian matrix than with the standard approach that uses Ω=1. We present calculations for model systems of identical bosons with harmonic and double-well potentials. Our results show promising potential for diminishing the computational cost of numerical simulations of various systems of trapped ultracold atoms.

Use of a non-relativistic basis for describing the low energy meson spectrum

We justify the use of harmonic oscillator eigenfunctions for expanding the fermionic fields of an effective field theory and evaluate its utility for solving a Hamiltonian inspired by the QCD formalism in the Coulomb Gauge. Since the functions involved in such expansion are non-relativistic, the Talmi-Moshinsky transformations can be used to recover the translational invariance of the center of mass of the mesonic states. Finally, many-body methods and an a posteriori flavor mixing procedure are used to compute a preliminary spectrum for the mesons below 1 GeV.

Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

We study the scaling asymptotics of the eigenspace projection kernels $\Pi_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 \Delta + |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $\Pi_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as $\hbar \to 0$. In previous work we proved that the density of zeros of Gaussian random eigenfunctions of $\hat{H}_{\hbar}$ have different orders in the Planck constant $\hbar$ in the allowed and forbidden regions: In the allowed region the density is of order $\hbar^{-1}$ while it is $\hbar^{-1/2}$ in the forbidden region. Our main result on nodal sets is that the density of zeros is of order $\hbar^{-\frac{2}{3}}$ in an $\hbar^{\frac{2}{3}}$-tube around the caustic. This tube radius is the `critical radius'. For annuli of larger inner and outer radii $\hbar^{\alpha}$ with $0< \alpha < \frac{2}{3}$ we obtain density results which interpolate between this critical radius result and our prior ones in the allowed and forbidden region. We also show that the Hausdorff $(d-2)$-dimensional measure of the intersection of the nodal set with the caustic is of order $\hbar^{- \frac{2}{3}}$.

Gauge angle dependence in time-dependent Hartree-Fock-Bogoliubov calculations ofO20+O20head-on collisions with the Gogny interaction

A numerical method to solve the TDHFB equations by using a hybrid basis of the two-dimensional harmonic oscillator eigenfunctions and one-dimensional Lagrange mesh with the Gogny effective interaction is applied to the head-on collisions of the superfluid nuclei ${}^{20}$O's. Taking the energies around the barrier top energy, the trajectories, pairing energies, and numbers of transferred nucleons are displayed. Their dependence on the relative gauge angle at the initial time is studied by taking typical sample points of the gauge angle. It turned out that the functional form of the flux of the neutrons across a section plane is proportional to the sine of the two times of the gauge angle.

The Berry–Keating Hamiltonian and the local Riemann hypothesis

The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have a real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry–Keating Hamiltonian projected onto the subspace of oscillator eigenfunctions of a lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert–Pólya conjecture. The p-adic case is also discussed.

A new basis set for molecular bending degrees of freedom

We present a new basis set as an alternative to Legendre polynomials for the variational treatment of bending vibrational degrees of freedom in order to highly reduce the number of basis functions. This basis set is inspired from the harmonic oscillator eigenfunctions but is defined for a bending angle in the range theta in [0:pi]. The aim is to bring the basis functions closer to the final (ro)vibronic wave functions nature. Our methodology is extended to complicated potential energy surfaces, such as quasilinearity or multiequilibrium geometries, by using several free parameters in the basis functions. These parameters allow several density maxima, linear or not, around which the basis functions will be mainly located. Divergences at linearity in integral computations are resolved as generalized Legendre polynomials. All integral computations required for the evaluation of molecular Hamiltonian matrix elements are given for both discrete variable representation and finite basis representation. Convergence tests for the low energy vibronic states of HCCH(++), HCCH(+), and HCCS are presented.

Effective harmonic oscillator description of anharmonic molecular vibrations

The validity of an effective harmonic oscillator approximation for anharmonic molecular vibrations is tested and compared with vibrational self consistent field and vibrational configurational interaction results. The effective harmonic oscillator is constructed variationally, by taking the trial wave function as a harmonic oscillator eigenfunction with the centroid and width parameter as variational paraeters. It is found that the effective harmonic oscillator approximation provides a description of the anharmonic eigenstates very similar to the vibrational self consistent field results. Coriolis coupling is also included in these studies.

Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions

We give a thorough analysis of the convergence properties of the configuration-interaction method as applied to parabolic quantum dots among other systems, including \emph{a priori} error estimates. The method converges slowly in general, and in order to overcome this, we propose to use an effective two-body interaction well-known from nuclear physics. Through numerical experiments we demonstrate a significant increase in accuracy of the configuration interaction method.

GROUND STATE AND LOW-EXCITED STATES OF A BOSE GAS IN A FINITE ANISOTROPIC TRAP

The motivation in this paper is to simulate numerically some properties of an interacting Bose–Einstein condensate at zero temperature in an axial symmetry trapping potential with finite amplitude for modeling the practical experimental cases. By use of the basis expansion using three-dimensional harmonic oscillator eigenfunctions, we obtain the ground-state wavefunction and the collective excitation spectra of the system in both usual harmonic potential and different amplitudes of the finite potential. After comparing our results for the finite potential with the data derived from the harmonic potential, we conclude that the finite trap in the practical experiments decreases the entire excitation frequencies in the whole regimes. This decrease is consistent with our analytic prediction qualitatively and agrees well with the experimental data quantitatively.

The optimized Rayleigh–Ritz scheme for determining the quantum-mechanical spectrum

The convergence of the Rayleigh–Ritz method with nonlinear parameters optimized through minimization of the trace of the truncated matrix is demonstrated by a comparison with analytically known eigenstates of various quasi-solvable systems. We show that the basis of the harmonic oscillator eigenfunctions with optimized frequency Ω enables determination of bound-state energies of one-dimensional oscillators to an arbitrary accuracy, even in the case of highly anharmonic multi-well potentials. The same is true in the spherically symmetric case of , if k > 0. For spiked oscillators with k < −1, the basis of the pseudoharmonic oscillator eigenfunctions with two parameters Ω and γ is more suitable, and optimization of the latter appears crucial for a precise determination of the spectrum.

Quantum theory of three-junction flux qubit with non-negligible loop inductance: Towards scalability

The three-junction flux qubit (quantum bit) consists of three Josephson junctions connected in series on a superconducting loop. We present a numerical treatment of this device for the general case in which the ratio ${\ensuremath{\beta}}_{Q}$ of the geometrical inductance of the loop to the kinetic inductance of the Josephson junctions is not necessarily negligible. Relatively large geometric inductances allow the flux through each qubit to be controlled independently with on-chip bias lines, an essential consideration for scalability. We derive the three-dimensional potential in terms of the macroscopic degrees of freedom, and include the possible effects of asymmetry among the junctions and of stray capacitance associated with them. To find solutions of the Hamiltonian, we use basis functions consisting of the product of two plane wave states and a harmonic oscillator eigenfunction to compute the energy levels and eigenfunctions of the qubit numerically. We present calculated energy levels for the relevant range of ${\ensuremath{\beta}}_{Q}$. As ${\ensuremath{\beta}}_{Q}$ is increased beyond $0.5$, the tunnel splitting between the ground and first excited states decreases rapidly, and the device becomes progressively less useful as a qubit.

Fock space construction of the quantum thermal flux operator

The construction of the flux operator in the Fock space of harmonic oscillator eigenfunctions is outlined. In one dimension, the flux operator is found to be F=( iω/√π) exp[− a +2/√2] [|1><0|−|0><1|] exp[− a 2/2]. Two approaches for the temperature propagation are discussed, and the thermal flux eigenvalues for a double well potential are presented.