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A four-parameter potential is analyzed, which contains the three-dimensional harmonic oscillator as a special case. This potential is exactly solvable and retains several characteristics of the harmonic oscillator, and also of the Coulomb problem. The possibility of similar generalizations of other potentials is also pointed out.
A finite difference method for the solution of the transonic flow about a harmonically oscillating wing is presented. The partial differential equation for the unsteady transonic flow was linearized by dividing the flow into separate steady and unsteady perturbation velocity potentials and by assuming small amplitudes of harmonic oscillation. The resulting linear differential equation is of mixed type, being elliptic or hyperbolic whereever the steady flow equation is elliptic or hyperbolic. Central differences were used for all derivatives except at supersonic points where backward differencing was used for the streamwise direction. Detailed formulas and procedures are described in sufficient detail for programming on high speed computers. To test the method, the problem of the oscillating flap on a NACA 64A006 airfoil was programmed. The numerical procedure was found to be stable and convergent even in regions of local supersonic flow with shocks.
In this work, we present the inverted harmonic oscillator (IHO) Hamiltonian as a paradigm to understand the quantum mechanics of scattering and time-decay in a diverse set of physical systems. As one of the generators of area preserving transformations, the IHO Hamiltonian can be studied as a dilatation generator, squeeze generator, a Lorentz boost generator, or a scattering potential. In establishing these different forms, we demonstrate the physics of the IHO that underlies phenomena as disparate as the Hawking–Unruh effect and scattering in the lowest Landau level (LLL) in quantum Hall systems. We derive the emergence of the IHO Hamiltonian in the LLL in a gauge invariant way and show its exact parallels with the Rindler Hamiltonian that describes quantum mechanics near event horizons. This approach of studying distinct physical systems with symmetries described by isomorphic Lie algebras through the emergent IHO Hamiltonian enables us to reinterpret geometric response in the lowest Landau level in terms of relativistic effects such as Wigner rotation. Further, the analytic scattering matrix of the IHO points to the existence of quasinormal modes (QNMs) in the spectrum, which have quantized time-decay rates. We present a way to access these QNMs through wave packet scattering, thus proposing a novel effect in quantum Hall point contact geometries that parallels those found in black holes.
The Schrodinger equation is used to exactly evaluate the propagator, wave function, energy expectation values, uncertainty values, and coherent state for a harmonic oscillator with a time dependent frequency and an external driving time dependent force. These quantities represent the solution of the classical equation of motion for the time dependent harmonic oscillator.
We discuss the semimicroscopic algebraic cluster model introduced recently, in which the internal structure of the nuclear clusters is described by the harmonic oscillator shell model, while their relative motion is accounted for by the Vibron model. The algebraic formulation of the model makes extensive use of techniques associated with harmonic oscillators and their symmetry group, SU(3). The model is applied to some cluster systems and is found to reproduce important characteristics of nuclei in the sd-shell region. An approximate SU(3) dynamical symmetry is also found to hold for the C-12 + C-12 system.
In a tokamak plasma, we measure nonlinear coupling between harmonics comprising a saturated edge oscillation, as predicted by nonlinear magnetohydrodynamic simulations. The coherent structure formed by this coupling suppresses bursty edge-localized modes that cause energy and particle loss. We also measure nonlinear coupling of the edge harmonic oscillation to core-resonant magnetic tearing modes. When tearing modes are present, the edge oscillation harmonics decouple, and edge-localized modes return. In this article, we conclude that nonlinear interaction with tearing modes interrupts the formation of the edge harmonic oscillation.
Singular, nonlinear, and Liapunov unstable equations are made regular and linear through transformations that change the perturbed planar problem of two bodies into unperturbed and undamped harmonic oscillators with constant coefficients, so that the stable solution may be immediately written in terms of the new variables. The use of arbitrary and special functions for the transformations allows the systematic discussion of previously introduced and novel anomalies. For the case of the unperturbed two-body problem, it is proved that if transformations are power functions of the radial variable, only the eccentric and the true anomalies (with the corresponding transformations of the radial variable) will result in harmonic oscillators. The present method significantly reduces computation requirements in autonomous space operations.
The nonrelativistic quark model is applied to heavy (nonrelativistic) meson (two-body) systems to obtain sufficiently accurate predictions of the spin-averaged mass levels of the charmonium and bottomonium spectra as an example of the three-dimensional harmonic oscillator. The present calculations do not include any spin dependence, but rather, mass values are averaged for different spins. Results for a charmed quark mass value of 1500 MeV/c-squared show that the simple harmonic oscillator model provides good agreement with experimental values for 3P states, and adequate agreement for the 3S1 states.
A formulation is given for a collection of phonons (sound) in a fluid at a non-zero temperature which uses the simple harmonic oscillator twice; one to give a stochastic thermal 'noise' process and the other which generates a coherent Glauber state of phonons. Simple thermodynamic observables are calculated and the acoustic two point function, 'contrast' is presented. The role of 'coherence' in an equilibrium system is clarified by these results and the simple harmonic oscillator is a key structure in both the formulation and the calculations.
It is shown that the system of two coupled harmonic oscillators possesses many interesting symmetries. It is noted that the symmetry of a single oscillator is that of the three-parameter group Sp(2). Thus two uncoupled oscillator exhibits a direct product of two Sp(2) groups, with six parameters. The coupling can be achieved through a rotation in the two-dimensional space of two oscillator coordinates. The closure of the commutation relations for the generators leads to the ten-parameter group Sp(4) which is locally isomorphic to the deSitter group O(3,2).
The harmonic oscillator with dissipation is studied within the framework of the Lindblad theory for open quantum systems. By using the Wang-Uhlenbeck method, the Fokker-Planck equation, obtained from the master equation for the density operator, is solved for the Wigner distribution function, subject to either the Gaussian type or the delta-function type of initial conditions. The obtained Wigner functions are two-dimensional Gaussians with different widths. Then a closed expression for the density operator is extracted. The entropy of the system is subsequently calculated and its temporal behavior shows that this quantity relaxes to its equilibrium value.
Vibrational heat-bath configuration interaction (VHCI)—a selected configuration interaction technique for vibrational structure theory—has recently been developed in two independent works [J. H. Fetherolf and T. C. Berkelbach, J. Chem. Phys. 154, 074104 (2021); A. U. Bhatty and K. R. Brorsen, Mol. Phys. 119, e1936250 (2021)], where it was shown to provide accuracy on par with the most accurate vibrational structure methods with a low computational cost. Here, we eliminate the memory bottleneck of the second-order perturbation theory correction using the same (semi)stochastic approach developed previously for electronic structure theory. This allows us to treat, in an unbiased manner, much larger perturbative spaces, which are necessary for high accuracy in large systems. Stochastic errors are easily controlled to be less than 1 cm−1. We also report two other developments: (i) we propose a new heat-bath criterion and an associated exact implicit sorting algorithm for potential energy surfaces expressible as a sum of products of one-dimensional potentials; (ii) we formulate VHCI to use a vibrational self-consistent field (VSCF) reference, as opposed to the harmonic oscillator reference configuration used in previous reports. Our tests are done with quartic and sextic force fields, for which we find that with VSCF, the minor improvements to accuracy are outweighed by the higher computational cost associated the matrix element evaluations. We expect VSCF-based VHCI to be important for more general potential representations, for which the harmonic oscillator basis function integrals are no longer analytic.
The dynamical properties of the wave and particle aspects of the harmonic oscillator can be studied with the help of the time-dependent Schroedinger equation (SE). Especially the time-dependence of maximum and width of Gaussian wave packet solutions allow to show the evolution and connections of those two complementary aspects. The investigation of the relations between the equations describing wave and particle aspects leads to an alternative description of the considered systems. This can be achieved by means of a Newtonian equation for a complex variable in connection with a conservation law for a nonclassical angular momentum-type quantity. With the help of this complex variable, it is also possible to develop a Hamiltonian formalism for the wave aspect contained in the SE, which allows to describe the dynamics of the position and momentum uncertainties. In this case the Hamiltonian function is equivalent to the difference between the mean value of the Hamiltonian operator and the classical Hamiltonian function.
A finite difference method to solve the unsteady transonic flow about harmonically oscillating wings was investigated. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady differential equation for small disturbances. The differential equation for the unsteady velocity potential is linear with spatially varying coefficients and with the time variable eliminated by assuming harmonic motion. An alternating direction implicit procedure was investigated, and a pilot program was developed for both two and three dimensional wings. This program provides a relatively efficient relaxation solution without previously encountered solution instability problems. Pressure distributions for two rectangular wings are calculated. Conjugate gradient techniques were developed for the asymmetric, indefinite problem. The conjugate gradient procedure is evaluated for applications to the unsteady transonic problem. Different equations for the alternating direction procedure are derived using a coordinate transformation for swept and tapered wing planforms. Pressure distributions for swept, untaped wings of vanishing thickness are correlated with linear results for sweep angles up to 45 degrees.
Previously, ELM-free and inter-ELM divertor peak heat flux reduction induced by an edge harmonic oscillation (EHO) was observed in NSTX (K.F. Gan et al., 2017 Nucl. Fusion 57 126053). This paper introduces new analysis of the EHO impact on the divertor heat flux. It was found that enhanced edge turbulence significantly altered the divertor heat flux footprint in NSTX H-modes, relative to ELM-free discharges. When the background edge fluctuation level was low, the EHO significantly increased the heat flux width and decreased the divertor peak heat flux. When the background edge fluctuation level was high, the EHO actually increased the divertor peak heat flux. As a result, it was also found that the heat flux width increased with increasing frequency of the EHO.
We test an isospectral potential from harmonic oscillator simulating H-bond interaction in DNA macromolecules.
Measuring systems to determine the real component of the complex amplitude of a harmonic oscillator are described. This amplitude is constant in the absence of driving forces, and the uncertainty principle accounts for the fact that only the real component can be measured precisely and continuously ('quantum nondemolition measurement'). Application of the measuring systems to the detection of gravitational waves is considered.
Although the problem of electromagnetic radiation by a quantum harmonic oscillator is considered in textbooks on quantum mechanics, some of its aspects have remained unclear until now. By this, we mean that usually the initial quantum states of both the oscillator and the field are assumed to be characterized by a definite energy level of the oscillator and definite occupation numbers of the field modes. In connection with growing interest in squeezed states, it would be interesting to analyze the general case when the initial states of both subsystems are arbitrary superpositions of energy eigenstates. This problem was considered in other work, where the power of the spontaneous emission was calculated in the case of an arbitrary oscillator's initial state, but the field was initially in a vacuum state. In the present article, we calculate the rate of the oscillator average energy, squeezing, and correlation parameter change under the influence of an arbitrary external radiation field. Some other problems relating to the interaction between quantum particles (atoms) or oscillators where the electromagnetic radiation is an arbitrary (in particular squeezed) state were investigated.