Harmonic Oscillator Energy Research Articles

Quantum effects in a free-standing graphene lattice: Path-integral against classical Monte Carlo simulations

In order to study quantum effects in a two-dimensional crystal lattice of a free-standing monolayer graphene, we have performed both path-integral Monte Carlo (PIMC) and classical Monte Carlo (MC) simulations for temperatures up to 2000 K. The REBO potential is used for the interatomic interaction. The total energy, interatomic distance, root-mean-square displacement of the atom vibrations, and the free energy of the graphene layer are calculated. The obtained lattice vibrational energy per atom from the classical MC simulation is very close to the energy of a three-dimensional harmonic oscillator $3{k}_{B}T$. The PIMC simulation shows that quantum effects due to zero-point vibrations are significant for temperatures $Tl1000$ K. The quantum contribution to the lattice vibrational energy becomes larger than that of the classical lattice for $Tl400$ K. The lattice expansion due to the zero-point motion causes an increase of 0.53% in the lattice parameter. A minimum in the lattice parameter appears at $T\ensuremath{\simeq}500$ K. Quantum effects on the atomic vibration amplitude of the graphene lattice and its free energy are investigated.

Quantum and thermal fluctuations in the harmonic chain and experimental implications

The nonzero ground-state energy of the quantum mechanical harmonic oscillator implies quantum fluctuations around the minimum of the potential with the mean-square value proportional to Planck's constant. On the other hand, thermal fluctuations occur when the oscillator is coupled to a heat bath. Using quantum statistical mechanics, this paper describes the transition from pure quantum fluctuations at zero temperature to classical thermal fluctuations in the high-temperature limit. It was early pointed out by Peierls that the classical mean-square thermal fluctuations in a harmonic chain—a linear chain of coupled harmonic oscillators—increase linearly with the distance of the masses from the fixed end of the chain, destroying long-range crystalline order. As shown here, the corresponding quantum fluctuations lead to a much slower logarithmic increase with the distance from the end. It is also shown that this behavior implies, for example, the absence of sharp Bragg peaks in x-ray scattering in an infinite chain at zero temperature, which instead shows power-law behavior typical for one-dimensional quantum liquids (called Luttinger liquids).

HOTB: High precision parallel code for calculation of four-particle harmonic oscillator transformation brackets

HOTB: High precision parallel code for calculation of four-particle harmonic oscillator transformation brackets

Isotope effects and spectroscopic assignments in the non-dissociative photoionization spectrum of N₂.

Photoionization efficiency spectra of (14)N2, (15)N(14)N, and (15)N2 from 15.5 to 18.9 eV were measured using synchrotron radiation at the Advanced Light Source at Lawrence Berkeley National Laboratory with a resolution of 6 meV, and significant changes in peak energies and intensities upon isotopic substitution were observed. Previously, we reported the isotope shifts and their applications to Titan's atmosphere. Here, we report more extensive experimental details and tabulate the isotope shifts of many transitions in the N2 spectrum, including those for (15)N(14)N, which have not been previously reported. The isotope shifts are used to address several long-standing ambiguities in spectral peak assignments just above the ionization threshold of N2. The feature at 15.677 eV (the so-called second "cathedral" peak) is of particular interest in this respect. The measured isotope shifts for this peak relative to (14)N2 are 0.015 ± 0.001 eV for (15)N2 and 0.008 ± 0.001 eV for (15)N(14)N, which match most closely with the isotope shifts predicted for transitions to the (A (2)Πu v' = 2)4sσ(g) (1)Π(u) state using Herzberg equations for the isotopic differences in harmonic oscillator energy levels plus the first anharmonic correction of 0.0143 eV for (15)N2 and 0.0071 eV for (15)N(14)N. More generally, the isotope shifts measured for both (15)N2 and (15)N(14)N relative to (14)N2 provide new benchmarks for theoretical calculations of interferences between direct and indirect autoionization states which can interact to produce intricate resonant structures in molecular photoionization spectra in regions near ionization thresholds.

The Kuo–Brown effective interaction: From 18O to the Sn isotopes

Abstract After briefly reviewing the pioneering work on effective interactions by Gerry Brown and his group, and the developments which followed, we apply present-day effective interactions to large-scale shell-model calculations on the entire range of Sn isotopes from 102Sn to 132Sn. We have made explorative calculations starting from three different nucleon–nucleon potentials (Argonne V18, CD-Bonn, and N3LO) and evaluated the higher-order contributions to the effective interaction from both G-matrix and V lowk interactions. Further, we have checked the convergence of intermediate-state excitations up to 10 ħ ω harmonic oscillator energy. Final extensive calculations were made of binding energies, excitation energies and B(E2) transition rates using an effective interaction based on a G-matrix evaluated from the chiral N3LO potential and including intermediate excitations up to 10 ħ ω harmonic oscillator energy. The energy spectra are well reproduced throughout the region while overbinding of the ground states emerges as valence nucleons are added. The B(E2) rates agree well for the heavy isotopes, while they seem too low for the lighter ones.

Formation and decay of Bose-Einstein condensates in an excited band of a double-well optical lattice

We study the formation and collision-aided decay of an ultra-cold atomic Bose-Einstein condensate in the first excited band of a double-well 2D-optical lattice with weak harmonic confinement in the perpendicular $z$ direction. This lattice geometry is based on an experiment by Wirth et al. The double well is asymmetric, with the local ground state in the shallow well nearly degenerate with the first excited state of the adjacent deep well. We compare the band structure obtained from a tight-binding (TB) model with that obtained numerically using a plane wave basis. We find the TB model to be in quantitative agreement for the lowest two bands, qualitative for next two bands, and inadequate for even higher bands. The band widths of the excited bands are much larger than the harmonic oscillator energy spacing in the $z$ direction. We then study the thermodynamics of a non-interacting Bose gas in the first excited band. We estimate the condensate fraction and critical temperature, $T_c$, as functions of lattice parameters. For typical atom numbers, the critical energy $k_BT_c$, with $k_B$ the Boltzmann constant, is larger than the excited band widths and harmonic oscillator energy. Using conservation of total energy and atom number, we show that the temperature increases after the lattice transformation. Finally, we estimate the time scale for a two-body collision-aided decay of the condensate as a function of lattice parameters. The decay involves two processes, the dominant one in which both colliding atoms decay to the ground band, and the second involving excitation of one atom to a higher band. For this estimate, we have used TB wave functions for the lowest four bands, and numerical estimates for higher bands. The decay rate rapidly increases with lattice depth, but stays smaller than the tunneling rate between the $s$ and $p$ orbitals in adjacent wells.

Harmonic oscillator with minimal length, minimal momentum, and maximal momentum uncertainties in SUSYQM framework

We consider a Generalized Uncertainty Principle (GUP) framework which predicts a maximal uncertainty in momentum and minimal uncertainties both in position and momentum. We apply supersymmetric quantum mechanics method and the shape invariance condition to obtain the exact harmonic oscillator eigenvalues in this GUP context. We find the supersymmetric partner Hamiltonians and show that the harmonic oscillator belongs to a hierarchy of Hamiltonians with a shift in momentum representation and different masses and frequencies. We also study the effect of a uniform electric field on the harmonic oscillator energy spectrum in this setup.

Extrapolation uncertainties in the importance-truncated no-core shell model

Background: The importance-truncated no-core shell model (IT-NCSM) has recently been shown to extend theoretical nuclear structure calculations of $p$-shell nuclei to larger model (${N}_{\mathrm{max}}$) spaces. The importance truncation procedure selects only relatively few of the many basis states present in a ``large'' ${N}_{\mathrm{max}}$ basis space, thus making the calculation tractable and reasonably quick to perform. Initial results indicate that the procedure agrees well with the NCSM, in which a complete basis is constructed for a given ${N}_{\mathrm{max}}$.Purpose: An analysis of uncertainties in IT-NCSM such as those generated from the extrapolations to the complete ${N}_{\mathrm{max}}$ space have not been fully discussed. We present a method for estimating the uncertainty when extrapolating to the complete ${N}_{\mathrm{max}}$ space and demonstrate the method by comparing extrapolated IT-NCSM to full NCSM calculations up to ${N}_{\mathrm{max}}=14$. Furthermore, we study the result of extrapolating IT-NCSM ground-state energies to ${N}_{\mathrm{max}}=\ensuremath{\infty}$ and compare the results to similarly extrapolated NCSM calculations. A procedure is formulated to assign uncertainties for ${N}_{\mathrm{max}}=\ensuremath{\infty}$ extrapolations.Method: We report on ${}^{6}\mathrm{Li}$ calculations performed with the IT-NCSM and compare them to full NCSM calculations. We employ the Entem and Machleidt chiral two-body next-to-next-to-next leading order (N3LO) interaction (regulated at 500 MeV/$c$), which has been modified to a phase-shift equivalent potential by the similarity renormalization group (SRG) procedure. We investigate the dependence of the procedure on the technique employed to extrapolate to the complete ${N}_{\mathrm{max}}$ space, the harmonic oscillator energy ($\ensuremath{\hbar}\ensuremath{\Omega}$), and investigate the dependence on the momentum-decoupling scale ($\ensuremath{\lambda}$) used in the SRG. We also investigate the use of one or several reference states from which the truncated basis is constructed.Results: We find that the uncertainties generated from various extrapolating functions used to extrapolate to the complete ${N}_{\mathrm{max}}$ space increase as ${N}_{\mathrm{max}}$ increases. The extrapolation uncertainties range from a few keV for the smallest ${N}_{\mathrm{max}}$ spaces to about 50 keV for the largest ${N}_{\mathrm{max}}$ spaces. We note that the difference between extrapolated IT-NCSM and NCSM ground-state energies, however, can be as large as 100--250 keV depending on the chosen harmonic oscillator energy ($\ensuremath{\hbar}\ensuremath{\Omega}$). IT-NCSM performs equally well for various SRG momentum-decoupling scales, $\ensuremath{\lambda}=2.02$ fm${}^{\ensuremath{-}1}$ and $\ensuremath{\lambda}=1.50$ fm${}^{\ensuremath{-}1}$.Conclusions: In the case of ${}^{6}$Li, when using the softened chiral nucleon-nucleon N3LO interaction, we have determined the difference between extrapolated ${N}_{\mathrm{max}}=\ensuremath{\infty}$ IT-NCSM and full NCSM calculations to be about 100--300 keV. As $\ensuremath{\hbar}\ensuremath{\Omega}$ increases, we find that the agreement with NCSM deteriorates, indicating that the procedure used to choose the basis states in IT-NCSM depends on $\ensuremath{\hbar}\ensuremath{\Omega}$. We also find that using multiple reference states leads to a better ground-state description than using only a single reference state.

On Aharonov–Casher bound states

In this work bound states for the Aharonov-Casher problem are considered. According to Hagen's work on the exact equivalence between spin-1/2 Aharonov-Bohm and Aharonov-Casher effects, is known that the $\boldsymbol{\nabla}\cdot\mathbf{E}$ term cannot be neglected in the Hamiltonian if the spin of particle is considered. This term leads to the existence of a singular potential at the origin. By modeling the problem by boundary conditions at the origin which arises by the self-adjoint extension of the Hamiltonian, we derive for the first time an expression for the bound state energy of the Aharonov-Casher problem. As an application, we consider the Aharonov-Casher plus a two-dimensional harmonic oscillator. We derive the expression for the harmonic oscillator energies and compare it with the expression obtained in the case without singularity. At the end, an approach for determination of the self-adjoint extension parameter is given. In our approach, the parameter is obtained essentially in terms of physics of the problem.

The effect of discontinuous wave functions on optimizations of VMC calculations of quantum systems

Here, a relatively good comparison has been done between the performance results of discontinuous and continuous forms of wave function on quantum systems as well. In this project, we’ve estimated the energy of ground and first state as the expectation value of Hamiltonian of harmonic oscillator and binding energy of nucleons confined in a Yukawa potential by making use of following three wave function: exponential, linear and Gaussian form which are convenient to optimize and may be evaluated rapidly are devised and tested and we stress the advantages of using expressions which are Gaussian in the variable parameters. Exponential function is quite different from a Gaussian ones as it has a cusp at x=0, hence a discontinuity in its derivatives. We then have performed to test inaccuracy of the discontinuous wave function and energy’s error of its exact value because using of exponential and linear function by mathematical statements of calculation of delta and Heaviside function and either VMC calculation of wave function optimization, as both of them will have the same result of error value. The result of our study is very promising for future applications of quantum Monte Carlo methods especially in many body systems

Construction of accelerating wavepackets

Accelerating wavepackets are obtained by applying the extended Galilean transformation to free-space particle wavepackets, or to time-dependent bound states. The transformed Schrödinger equation contains a potential term, which drives the acceleration. The method reproduces Darwin’s result for an accelerating Gaussian wavepacket in a constant force field, and Schrödinger’s oscillating wavepacket in a harmonic well. We also derive a new accelerating Airy wavepacket in a gravitational field, and give a new derivation of oscillating wavepackets based on the higher harmonic oscillator energy eigenstates. These results illustrate an analytic method of dealing with wavepackets accelerating in force fields.

Continuous Quantum Hypothesis Testing

I propose a general quantum hypothesis testing theory that enables one to test hypotheses about any aspect of a physical system, including its dynamics, based on a series of observations. For example, the hypotheses can be about the presence of a weak classical signal continuously coupled to a quantum sensor, or about competing quantum or classical models of the dynamics of a system. This generalization makes the theory useful for quantum detection and experimental tests of quantum mechanics in general. In the case of continuous measurements, the theory is significantly simplified to produce compact formulas for the likelihood ratio, the central quantity in statistical hypothesis testing. The likelihood ratio can then be computed efficiently in many cases of interest. Two potential applications of the theory, namely, quantum detection of a classical stochastic waveform and test of harmonic-oscillator energy quantization, are discussed.

Electronic spectra of ArXe molecules in the region of Xe* (5d, 7s, 7p, 6p′), 80 300–89 500 cm−1, using resonantly enhanced multiphoton ionization

The electronic spectra of ArXe molecules in the 80 300–89 500 cm−1 region were recorded by (2 + n) and (3 + n) REMPI methods. The vibrational progressions attributed to transitions of molecules from the ground state to the bounded excited state and wide unstructured bands related to transitions to the continuous upper state were obtained. The molecular constants of ArXe* were calculated for all the observed progressions in the 80 300–87 000 cm−1 region as an approximation of an anharmonic oscillator and the Morse potential. For different excited states the energy of harmonic oscillator and the dissociation energy are changed from 10 to 100 cm−1 and from 70 to 750 cm−1, respectively.

QUANTUM GRAVITATIONAL CORRECTIONS TO THE HYDROGEN ATOM AND HARMONIC OSCILLATOR

It is shown that the rate of corrections to the hydrogen atom and harmonic oscillator energy levels due to the profound quantum gravitational effect of space–time dimension running/reduction coincides well with those obtained by means of minimum-length-deformed quantum mechanics. The rate of corrections is pretty much the same within the accuracy with which we can judge the quantum gravitational corrections at all. Such a convergence of results makes the concept of space–time dimension running more appreciable. As a remarkable distinction, the energy shift due to dimension reduction has the opposite sign as compared with the correction obtained by means of minimum-length-modified quantum mechanics. Therefore, the sign of total quantum gravitational correction remains somewhat obscure.

Coulomb crystallites from harmonically confined charged bosons in two dimensions

We exploit rotational-symmetry breaking in the one-body density to examine the formation of structures insystems of N strongly coupled charged bosons with logarithmic repulsions inside isotropic two-dimensional harmonictraps, with N in the range from 2 to 7. The results serve as a map for ordered arrangementsof vortices in a trapped Bose–Einstein condensate. Two types ofN-body wavefunctions are assumed: (i) a permanent of N identical Gaussian orbitals centred at variationally determined sites, and (ii) a permanent of N orthogonal orbitals built from harmonic-oscillator energy eigenstates. With increasingcoupling strength, the bosons in the orbitals localize into polygonal-ringlike crystalline patterns (‘Wigner molecules’), whereasthe wavefunctions describe low energy excited states containing delocalized bosons as in supersolid crystallites (‘supermolecules’).For N = 2 at strong coupling both states describe a Wigner dimer.

Energy levels and the de Broglie relationship for high school students

In this article, four examples of possible lessons on energy levels for high school aredescribed: a particle in a box, a finite square well, the hydrogen atom and a harmonicoscillator. The energy levels are deduced through the use of the steady-state condition andthe de Broglie relationship. In particular, the harmonic oscillator energy levels are deducedusing correspondence with circular uniform motion.

Degeneracy of confined D‐dimensional harmonic oscillator

AbstractUsing the mathematical properties of the confluent hypergeometric functions, the conditions for the incidental, simultaneous, and interdimensional degeneracy of the confined D‐dimensional (D > 1) harmonic oscillator energy levels are derived, assuming that the isotropic confinement is defined by an infinite potential well and a finite radius Rc. Very accurate energy eigenvalues are obtained numerically by finding the roots of the confluent hypergeometric functions that confirm the degeneracy conditions. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

A THEOREM ON CYCLIC HARMONIC OSCILLATORS

It is proven that the energy of a quantum mechanical harmonic oscillator with a generically time-dependent but cyclic frequency, ω(t0) = ω(0), cannot decrease on an average if the system is originally in a stationary state, after the system goes through a full cycle. The energy exchange always takes place in the direction from the macroscopic system (environment) to the quantum microscopic system.

Bound states ofn-dimensional harmonic oscillator decorated with Dirac delta functions

Bound state solutions of the Schrodinger equation have been investigated for n-dimensional (n ≥ 2) harmonic oscillator potential decorated with any finite number (P) of Dirac delta functions. The potential is radially symmetric and given as , where σis are arbitrary real numbers, r1 < r2 < ⋅⋅⋅ < rP and ri (0, + ∞). We have demonstrated that addition of Dirac delta functions lifts the accidental degeneracies of n-dimensional harmonic oscillator energy levels and leaves only the degeneracy due to the radial symmetry. Explicit forms of bound state eigenfunctions and the eigenvalue equation are given for n, l values, where n is the space dimension and l is the degree of n-dimensional spherical harmonics. We have shown that, for given n and l, there are a countably infinite number of bound state energy levels which are continuous functions of ω, σis and at most P of them can be negative.

The space dent of sphere-symmetry harmonic oscillator in ground state

A rigorous solution of Schrdinger equation of an spheresymmetry harmonic oscillator is studied by the power series method.We found that wave function ψ(r,θ,φ)|r～0 can be infinite, rψ(r,θ,φ)|r～0 must be finite; this is in accord with Born's statistical explanation. The groundstate energy of the spheresymmetry harmonic oscillator is ω/2. The space dent will appear in the ground state of the spheresymmetry harmonic oscillator.