Harmonic Oscillator Eigenstates Research Articles

Angular momentum eigenstates of the isotropic 3-D harmonic oscillator: Phase-space distributions and coalescence probabilities

The isotropic 3-dimensional harmonic oscillator potential can serve as an approximate description of many systems in atomic, solid state, nuclear, and particle physics. In particular, the question of 2 particles binding (or coalescing) into angular momentum eigenstates in such a potential has interesting applications. We compute the probabilities for coalescence of two distinguishable, non-relativistic particles into such a bound state, where the initial particles are represented by generic wave packets of given average positions and momenta. We use a phase-space formulation and hence need the Wigner distribution functions of angular momentum eigenstates in isotropic 3-dimensional harmonic oscillators. These distribution functions have been discussed in the literature before but we utilize an alternative approach to obtain these functions. Along the way, we derive a general formula that expands angular momentum eigenstates in terms of products of 1-dimensional harmonic oscillator eigenstates.

Arbitrary-order even and odd winking states of excited Schrödinger's cats

We consider how the higher-energy excited Schrödinger's cats can be transformed into the even and odd winking states, the main feature of which consists in the periodic jumping of the probability density distribution from one to other regions of space. In particular, the winking states can be formed among closely displaced ground and arbitrary-order coherent states involved into the Schrödinger's “alive” and “dead” cat dynamics. From a pedagogical standpoint it is worth mentioning that the general analytical solution for arbitrary-order winking states can be obtained practically without any calculations. We use only the method of mathematical induction and the “often forgotten” quantum-mechanical Husimi and Taniuti symmetry transformation. We show that the Husimi and Taniuti gauge invariance establishes a one to one correspondence between the harmonic oscillator eigenstates and the higher-energy coherent states of the same order in such a way that their space and time-dependent phase factors are identical for all orders of coherent states. By way of illustration, we exemplify the main logical-operation properties of winking states up to the fifth-order.

FACt: FORTRAN toolbox for calculating fluctuations in atomic condensates

We develop a FORTRAN code to compute fluctuations in atomic condensates (FACt) by solving the Bogoliubov–de Gennes (BdG) equations for two component Bose–Einstein condensate (TBEC) in quasi-two dimensions. The BdG equations are recast as matrix equations and solved self consistently. The code is suitable for handling quantum fluctuations as well as thermal fluctuations at temperatures below the critical point of Bose–Einstein condensation. The code is versatile, and the ground state density profile and low energy excitation modes obtained from the code can be easily adapted to compute different properties of TBECs — ground state energy, overlap integral, quasi particle amplitudes of BdG spectrum, dispersion relation and structure factor and other related experimental observables. Program summaryProgram Title: FACtProgram Files doi:http://dx.doi.org/10.17632/5h348ndydg.1Licensing provisions: MITProgramming language: FORTRAN 90External routines/libraries: ARPACKNature of problem: Compute the ground state density profile, ground state energy and chemical potential for individual species, evaluate the quasiparticle mode energies and corresponding amplitudes which can capture the transformation of the modes against the change of the parameters (intraspecies interaction, interspecies interaction, anisotropy parameter etc.) using Hartree–Fock–Bogoliubov theory with the Popov approximation. Calculate the overlap integral, dispersion relation and structure factor.Solution method: In the first step, the pair of coupled Gross–Pitaevskii equations (CGPEs) is solved using split time-step Fourier pseudospectral method to compute the condensate density. To solve the BdG equations, as a basic input the first Nb harmonic oscillator eigenstates are chosen as a basis to generate the BdG 011 matrix with dimension of 4(Nb+1)×4(Nb+1). Since the matrix size rapidly increases with Nb, Arpack routines are used to diagonalize the BdG matrix efficiently. To compute the fluctuation and non-condensate density, a set of the low energy quasiparticle amplitudes above a threshold value of the Bose factor are considered. The equations are then solved iteratively till the condensate, and non-condensate densities converge to predefined accuracies. To accelerate the convergence we use the method of successive under-relaxation (SUR).Additional comments including restrictions and unusual features: For a large system size, if the harmonic oscillator basis size is also taken to be large, the dimension of the BdG matrix becomes huge. It may take several days to compute the low energy modes at finite temperature and this package may be computationally expensive.After successful computation of this package, one should obtain the equilibrium density profiles for TBEC, low energy Bogoliubov modes and the corresponding quasiparticle amplitudes. In addition, one can calculate the dispersion relation, structure factor, overlap integral, correlation function, etc. using this package with minimal modifications. In the theory section of the manuscript, we have provided the expressions to compute the above quantities numerically.

Clock Time in Quantum Cosmology

We consider the conditioning of the timeless solution to the Wheeler–DeWitt equation by a predefined matter clock state in the simple scenario of an FLRW universe. The resulting evolution of the geometrodynamical degree of freedom with respect to clock time is characterized by the “Berry connection” of the reduced geometrodynamical space, which relies on the coupling of the clock with the geometry. When the connection vanishes, the standard Schrödinger equation is obtained for the geometry with respect to clock time. When one considers environment-induced decoherence in the semiclassical limit, this condition is satisfied, and clock time coincides with cosmic time. Explicit results for the conditioned wave functions for minimal clocks made up of two quantum harmonic oscillator eigenstates are shown.

Synthetic dimensions for cold atoms from shaking a harmonic trap

We introduce a simple scheme to implement synthetic dimensions in ultracold atomic gases, which only requires two basic and ubiquitous ingredients: the harmonic trap, which confines the atoms, combined with a periodic shaking. In our approach, standard harmonic oscillator eigenstates are reinterpreted as lattice sites along a synthetic dimension, while the coupling between these lattice sites is controlled by the applied time-modulation. The phase of this modulation enters as a complex hopping phase, leading straightforwardly to an artificial magnetic field upon adding a second dimension. We show that this artificial gauge field has important consequences, such as the counterintuitive reduction of average energy under resonant driving, or the realisation of quantum Hall physics. Our approach offers significant advantages over previous implementations of synthetic dimensions, providing an intriguing route towards higher-dimensional topological physics and strongly-correlated states.

A model study on a pair of trapped particles interacting with an arbitrary effective range

We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s-wave interactions over a wide range of s-wave scattering length as including the unitarity limits . Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch et al (1998 Foundations of Physics 28 549) are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in a 1D harmonic trap taking one dimensional finite-range model potential. We have also calculated the bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio η of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for . However, for a large range, one can almost get 1D results for smaller values of η. This study will be important for the exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.

Wigner phase space distribution via classical adiabatic switching.

Evaluation of the Wigner phase space density for systems of many degrees of freedom presents an extremely demanding task because of the oscillatory nature of the Fourier-type integral. We propose a simple and efficient, approximate procedure for generating the Wigner distribution that avoids the computational difficulties associated with the Wigner transform. Starting from a suitable zeroth-order Hamiltonian, for which the Wigner density is available (either analytically or numerically), the phase space distribution is propagated in time via classical trajectories, while the perturbation is gradually switched on. According to the classical adiabatic theorem, each trajectory maintains a constant action if the perturbation is switched on infinitely slowly. We show that the adiabatic switching procedure produces the exact Wigner density for harmonic oscillator eigenstates and also for eigenstates of anharmonic Hamiltonians within the Wentzel-Kramers-Brillouin (WKB) approximation. We generalize the approach to finite temperature by introducing a density rescaling factor that depends on the energy of each trajectory. Time-dependent properties are obtained simply by continuing the integration of each trajectory under the full target Hamiltonian. Further, by construction, the generated approximate Wigner distribution is invariant under classical propagation, and thus, thermodynamic properties are strictly preserved. Numerical tests on one-dimensional and dissipative systems indicate that the method produces results in very good agreement with those obtained by full quantum mechanical methods over a wide temperature range. The method is simple and efficient, as it requires no input besides the force fields required for classical trajectory integration, and is ideal for use in quasiclassical trajectory calculations.

The time-dependent quantum harmonic oscillator revisited: Applications to quantum field theory

In this article, we formulate the study of the unitary time evolution of systems consisting of an infinite number of uncoupled time-dependent harmonic oscillators in mathematically rigorous terms. We base this analysis on the theory of a single one-dimensional time-dependent oscillator, for which we first summarize some basic results concerning the unitary implementability of the dynamics. This is done by employing techniques different from those used so far to derive the Feynman propagator. In particular, we calculate the transition amplitudes for the usual harmonic oscillator eigenstates and define suitable semiclassical states for some physically relevant models. We then explore the possible extension of this study to infinite dimensional dynamical systems. Specifically, we construct Schrödinger functional representations in terms of appropriate probability spaces, analyze the unitarity of the time evolution, and probe the existence of semiclassical states for a wide range of physical systems, particularly, the well-known Minkowskian free scalar fields and Gowdy cosmological models.

Properties of Minimum Uncertainty Wavelets and Their Relations to the Harmonic Oscillator and the Coherent States

We consider additional aspects of the recently derived “minimum uncertainty” (μ) wavelets. In particular, we show that they are fundamentally related to both the harmonic oscillator eigenstates and the canonical coherent states that play a fundamental role in quantum dynamics. In addition, we derive new raising and lowering operators that apply to the μ-wavelets. Finally, we explore in some detail the senses in which the μ-wavelets form complete sets that can be used in a variety of applications in quantum dynamics.

Wavelets and orthogonal polynomials based on harmonic oscillator eigenstates

We present a new method for the construction of wavelets. The main tool is the harmonic oscillator eigenstates. With the help of their properties we are able to construct wavelets which have explicit expressions and have exponential decay in both time domain and frequency domain. Moreover we get a new family of polynomials which are orthogonal in a weighted squarely integrable function space.

Coherent states for the Kepler motion

The dynamics of the electron wave function in a hydrogen atom is mapped onto that of a two-dimensional harmonic oscillator. Based on the harmonic-oscillator eigenstates, a coherent state wave function for the atomic electron is proposed.

Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates

The problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one-dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential Vn(t)=−∫−∞∞(Hn(x))2 ln |x−t|e−x2 dx at the zeros of the Hermite polynomial Hn(x). Here, a closed analytical expression for Vn(t) is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of Hn(x) is known. An inequality for the values of Vn(t) at the zeros of Hn(x) is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for Vn(t) is written in an alternative way, which allows the entropies to be expressed in terms of the even-order spectral moments of the Hermite polynomials. The asymptotic (n≫1) limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically valid.

Application of U(2)⊃O(2) and U(2)⊃U(1) dynamical symmetries to diatomic molecular vibrations

Algebraic Hamiltonians based on the Casimir invariants of U(1) and O(2) dynamical symmetries of U(2) are used to describe the vibrational energy level spectrum of a diatomic molecule. The set of vibrational eigenstates arising from the U(1) chain remains closed under the operation of the appropriate ladder operators, whereas no such closure occurs in the case of the O(2) chain. This result is shown to be a consequence of the differing roles of the dimension of the irreducible representation of U(2), the so-called vibron number, in the two dynamical symmetry chains. A simple procedure is proposed to effect closure in the O(2) chain, as required for the eigenstates of a Morse oscillator Hamiltonian, to which the algebraic Hamiltonian based on the quadratic Casimir invariant of O(2) reduces within a coordinate representation. For each symmetry chain, application of the Holstein-Primakoff transformation leads to a set of ladder operators that act on one-dimensional harmonic oscillator eigenstates. Explicit comparison of the two dynamical symmetry chains is made with respect to the vibrational energy spectrum of the hydrogen molecule in its ground electronic state.

Classical limit of the harmonic oscillator Wigner functions in the Bargmann representation

Writing the Wigner functions of any pair of harmonic oscillator eigenstates in the Bargmann representation, a direct and detailed proof is given of their convergence (in the sense of distributions) to at the classical limit , , , m-n fixed, .

Coupled harmonic oscillators, generalized harmonic-oscillator eigenstates and coherent states

Coupled-harmonic-oscillator Hamiltonians, modelling different physical processes, are discussed along with their eigenfunctions, which are shown to be generalized harmonic-oscillator functions with many indices and variables. We introduce the relevant coherent states and analyse their peculiar properties. The physical and mathematical consequences of our study are finally considered.

Quantum harmonic oscillator in a strong electrostatic wave

We study the quantum behaviour of a harmonic oscillator that undergoes the effect of a strong wave. This is done using the strong-field approximation where the order 0 is driven only by the external perturbation. It is shown that the wave function is just a superposition of coherent states labelled by an integer number. In a first approximation, such coherent states can be taken orthogonal so that the probability of finding the system in one of them is just the square of the Bessel function of order given by the number labelling the state. In terms of harmonic oscillator eigenstates, transition amplitudes are significant just for high values of the level numbers in the same approximation of orthogonality of the coherent states.

Transformation to pseudo-SU (3) in heavy deformed nuclei

The explicit form of the unitary operator connecting the normal parity harmonic oscillator eigenstates with the full set of eigenstates of a pseudo-oscillator is determined. This operator is used to calculate the transformed Nilsson hamiltonian in pseudo- space. For the latter, the spin-orbit partners are almost degenerate showing that pseudo-SU (3) symmetry is approximately preserved in heavy nuclei.

Perturbational and variational treatments of the Morse oscillator

The Morse oscillator hamiltonian is expressed as an infinite expansion in powers of a natural perturbation parameter, the square root of the anharmonicity constant, relative to the simple harmonic oscillator as zeroth-order hamiltonian. A transformation of variables leads to a hamiltonian which involves terms no higher than second order in this natural perturbation parameter. In both cases, the exact bound state eigenvalues of the Morse oscillator are given by second-order perturbation theory. The Schrödinger equation corresponding to the transformed Morse hamiltonian is solved variationally, via a complete set expansion in simple harmonic oscillator eigenstates. Accurate approximations to the exact eigenvalues and eigenfunctions of bound states of the Morse oscillator can be obtained for all but the very highest levels.

Microscopic calculations in the A=6 system.

The A=6 structure problem is solved within the framework of the nuclear shell model. Model states are expanded upon a basis of properly symmetrized, translationally invariant harmonic oscillator eigenstates including states of up to 6\ensuremath{\Elzxh}\ensuremath{\omega} of excitation. The model interaction is based upon a modification of the two-body Sussex interaction. Eigenvalues and (${J}^{\ensuremath{\pi}}$,T) values are predicted for the ground states of the following systems: $^{6}\mathrm{n}$, $^{6}\mathrm{H}$, $^{6}\mathrm{He}$, $^{6}\mathrm{Li}$, $^{6}\mathrm{Be}$, $^{6}\mathrm{B}$, and $^{6}\mathrm{C}$. The ground state binding energies are within 4 percent of experiment for $^{6}\mathrm{He}$, $^{6}\mathrm{Li}$, and $^{6}\mathrm{Be}$. Excited states for the $^{6}\mathrm{He}$, $^{6}\mathrm{Li}$, and $^{6}\mathrm{Be}$ systems are determined and compared to experiment and other calculations. The model spectra for the $^{6}\mathrm{Li}$ system is similar to that proposed by Ajzenberg-Selove. However, the $^{6}\mathrm{He}$ and $^{6}\mathrm{Be}$ spectra contain levels in addition to those suggested by existing compilations. In particular, the $^{6}\mathrm{He}$ and $^{6}\mathrm{Be}$ level spectra have the following level ordering: (${0}^{+}$,1), (${2}^{+}$, 1), (${1}^{+}$,0), (${0}^{+}$,1), (${0}^{+}$,1), (${3}^{+}$,0) (${2}^{+}$,1), (${4}^{\mathrm{\ensuremath{-}}}$,1), (${2}^{\mathrm{\ensuremath{-}}}$,1), (${3}^{\mathrm{\ensuremath{-}}}$,1), and (${4}^{+}$,1).

Shell-model calculations in the A = 5 system

The A = 5 structure problem is solved within the framework of the nuclear shell model. The model states are expanded on a basis of properly symmetrized translationally invariant harmonic oscillator eigenstates including states of up to 5(kh)gw of oscillator excitation. Ground and excited levels of the 5H, 5He, 5Li and 5Be systems are presented. The two-body interaction is based on the Sussex matrix elements. Model results are in good agreement with measured A = 5 level positions. The model T = 3 2 results lead to a quartet of levels, whose positions are given roughly by the isobaric multiplet mass equation.