Analysis of energy distribution in a loaded quantum anharmonic oscillator in a wide temperature range

Analysis of the energy distribution in an ensemble of quantum anharmonic oscillators loaded by an external force in a wide temperature range (from T = 0) is carried out using a general approach based on the virial theorem. At T = 0, anharmonic effects are observed: a linear variation of zero-point energy of an oscillator under loading (energy decrease during extension and increase under compression) and a linear variation of the average kinetic and potential energy components. At high temperatures, at which the dynamics of the oscillators becomes classical, the anharmonic effects are manifested in a linear variation in the vibrational energy and a linear variation in the average kinetic and potential energy components upon an increase in force. Mutually compensating variation in the average kinetic and potential energy components of the internal dynamic energy of an oscillator (energy redistribution upon loading) takes place both at low and high temperatures.     

Energy features of a loaded quantum anharmonic oscillator

A quantum anharmonic oscillator in the ground state has been considered under the conditions of loading with an external force. The wave functions have been calculated for different forces, and the eigenvalues of the energy of the system have been determined as a function of the load. It has been established that the zero-point energy of the oscillator varies linearly with a variation in the force (decreases under tension and increases under compression) and that the average kinetic and potential components of the energy are also characterized by linear variations.     

Energy of the frequency-elastic effect in solids

A decrease in the frequency of skeletal vibrations (frequency-elastic effect) has been measured using Raman spectroscopy, and the stretching of backbone interatomic bonds in polyethylene molecules under elastic tensile loading of oriented polyethylene fibers has been measured using X-ray diffraction. It has been found that there are differences in the sign and magnitude of the changes in the zero-point energy and the work of the external force. The energy of the frequency-elastic effect has been explained in terms of the influence exerted by the initial (before loading) anharmonic stretching of backbone bonds and the force of anharmonic pressure, with the separation of the anharmonic (potential) component of the zero-point energy of the solid. A change in the frequency of vibrations corresponds to a change in the harmonic component of the zero-point energy. The loading with an external force causes a redistribution of the zero-point energy components. An energy analysis of the loaded quantum anharmonic oscillator has confirmed the conclusion regarding the mechanism of energy transfer and revealed that, under loading, there is a redistribution of the average values of the kinetic and potential components of the internal energy of the oscillator.     

Energy of the elastic loading of anharmonic solids

This paper reports on the results of measurements of changes in the temperature of a solid under the adiabatic elastic loading (thermoelastic effect), the coefficient of thermal expansion, and the Young’s modulus of a rigid-chain vitrified polymer, namely, polyimide. It has been found that there are differences in the sign and magnitude of the changes in the energy of thermal origin in samples and the work of the external force. The energy of the thermoelastic effect has been explained in terms of the influence exerted by the anharmonic expansion of a solid, with the separation of the quasi-static potential and dynamic components of the thermal energy of the solid. The loading with an external force causes a redistribution of the thermal energy. A change in the temperature of the solid corresponds to a change in the dynamic component. An energy analysis of the adiabatically loaded anharmonic oscillator has confirmed the conclusion regarding the mechanism of energy transfer and revealed that, under loading, there is a redistribution of the kinetic and potential components of the internal energy of the oscillator.     

Energetics of the thermoelastic effect in solids

The dependence of the temperature on the external adiabatic deformation is determined for a one-dimensional model of a solid — chains of atoms with an anharmonic interaction. The resulting dependences of the average kinetic and potential components of the internal energy on this deformation are compared with a model of adiabatic loading of a single oscillator. Fiz. Tverd. Tela (St. Petersburg) 40, 1548–1551 (August 1998)     

Energetics of an adiabatically-loaded excited anharmonic oscillator

The dependence of the energy characteristics on an adiabatically slowly increasing external force is determined analytically for an anharmonic oscillator. The analytical results are confirmed by a numerical calculation. The nature of the force dependences of the energy characteristics are determined and discussed. Fiz. Tverd. Tela (St. Petersburg) 39, 153–157 (January 1997)     

Simulation of thermoelastic properties of solids in the framework of an ensemble of anharmonic oscillators

It has been shown that, in a classical ensemble of anharmonic oscillators, the mean value of the oscillator coordinate is a classical parameter in the sense that the statistical sum of the ensemble satisfies, to the second order in the anharmonicity constant, the stationary condition with respect to this parameter. This stationary condition is equivalent to the classical condition for the balance of external and internal forces acting on the oscillator. This equivalence is justified by the fact that the statistical sum, which is stationary with respect to the mean oscillator coordinate, agrees within this accuracy with the usual statistical sum of independent anharmonic oscillators. After introducing the classical parameter into a large thermodynamic system, the energy balance under the mechanical deformation of the system is realized through the exchange between two scale levels: the energy of oscillations at the microlevel and the macroscopic potential energy of deformation of the sample as a whole.     

Harmonics generation and chaos in scattering of phonons from anharmonic surfaces

The effective equations of motion for a surface atom in an anharmonic surface potential have been derived for dispersionless one-dimensional substrates. The system is equivalent to a non-linear damped oscillator (Duffing oscillator) with the forcing term depending on the form of the incident wave. Efficiency of harmonics generation, phonon reflection coefficients, effective local density of states, regions of chaotic motion and windows of periodic motion have been comparatively evaluated for the system subject to an oscillating external force and to the irradiation by a monochromatic phonon coming from the bulk. Comparison of the resonant desorption of the surface atom within a given time interval has been made for the same example of anharmonic surface potential in both types of perturbation. Received 9 November 1998     

Energy exchange between vibration modes of a graphene nanoflake oscillator: Molecular dynamics study

We investigated the oscillatory behaviors of a square graphene-nanoflake (GNF) on a rectangular GNF via classical molecular dynamics simulations, and analyzed the energy exchange and the oscillation frequencies for three different modes. The simulation results using a model structure show that the GNF oscillator can be considered as a high frequency oscillator. As its initial velocity increases, its telescoping region increases, then its structural asymmetry along the axis due to own small rotation exerted asymmetric van der Waals (vdW) force on it, and finally, this asymmetric vdW force enhances its rotational motions during its axial translational motions. So the initial kinetic energy of the axial translational motion is changed into the energy of the orthogonal vibrational and the rotational motions. Its resonance frequencies are dependent on the aspect ratio of the bottom rectangular GNF, the difference between the lengths of the GNF oscillator and the bottom rectangular GNF, and the initial velocity.     

Dirac bound states of anharmonic oscillator in external fields

We explore the effect of the external magnetic and Aharonov–Bohm (AB) flux fields on the energy levels of Dirac particle subjects to mixed scalar and vector anharmonic oscillator field in the two-dimensional (2D) space. We calculate the exact energy eigenvalues and the corresponding un-normalized two-spinor-components wave functions in terms of the chemical potential parameter, magnetic field strength, AB flux field and magnetic quantum number by using the Nikiforov–Uvarov (NU) method.     

Hydrodynamics of resonance oscillations of columns of inelastic particles

We study oscillations of a one-dimensional (1D) column of N slightly inelastic particles, produced by a piston vibrating at one end of a closed tube. It is found that for large enough vibrational amplitudes of the piston, the column oscillates periodically with the period equal to the vibrational period. The oscillation patterns are governed by the shock waves propagating across the column. The averaged kinetic energy per particle is shown to be proportional to the square of the vibrational frequency, omega. This energy also strongly depends on the vibrational amplitude. The maximal value of this kinetic energy achievable by these external vibrations is found to be of order omega(2)L(2), where L is the total volume (length) of the tube free of particles. The above results on the column resonance oscillations are also predicted by a 3D hydrodynamic model of an inelastic granular gas.     

非谐振子势的精确解和双波函数描述

求解了非谐振子势V(x)=x²/2+g/2x²的本征方程，给出了精确的能谱方程和归一化波函数.应用双波函数理论，得到了在非谐振子势场中单粒子运动状态的力学量的时间演化方程. 关键词：     

Control of effective potential minima for Kapitza oscillator by periodical kicking pulses

Here we propose the modification of Kapitza procedure of averaging to control the effective energy minima of the oscillator driven by periodical external field. We demonstrate the effect of control with relatively low frequency of the field of oscillation to compare with sin- or cos-periodical force.     

Oscillating Propagation of Kink in Nondissipative Frenkel-Kontorova Chain Due to External DC Force

We report the oscillating propagation of kink in a nondissipative Frenkel-Kontorova （FK） chain driven by external DC force, which is different from the usual propagation of localized modes with equal speed. When the kink moves in the opposite direction of the external DC force, the kink will be accelerated and the potential of the FK chain in the external force field is transformed to be the kinetic energy of the kink. If the kink reaches the boundary of the FK chain, the kink will be bounced back and moves in the opposite direction, then the kink will be decelerated gradually and the kinetic energy of the kink ＆ transformed to be the potential of the FK chain in the external force field. If the speed of the kink reaches zero, the kink will move in the opposite direction again driven by the external DC force, and a new oscillating cycle begins. Simulation result demonstrates exactly the transformation between the kinetic energy of the kink and the potential of the FK chain in the external force field. The interesting energy exchange is induced by the special topology of kinks, and other localized modes, such as breathers and envelope solitons, have no the interesting phenomenon.     

Exact solutions for a classical doubly anharmonic oscillator

We obtain exact solutions for the motion of a classical anharmonic oscillator in the potential Bx ²–|A|x ⁴+Cx ⁶, and discuss the energy dependence of the frequencies of oscillation.     

Quantum corrected cell model for an anharmonic generalized Lennard-Jones solid

A simple method is proposed to dispose the quantum effect and anharmonic effect at the same time. Considering the quantum effect is remarkable only at low temperature, and tends to zero at high temperature, the potential energy of an atom is expanded harmonically to consider the quantum effect of solids within the harmonic oscillator framework. The anharmonic effect is remarkable only at high temperature, and tends to zero at low temperature, it was disposed by using a classical approximation. The universal formalism is applied to the generalized Lennard-Jones solid. The comparison shows that the results with and without anharmonic effect are in agreement with each other at some low temperature, to which the Einstein model is applicable. The results without anharmonic effect become divergent at slightly higher temperatures; however, the results including anharmonic effect are in good agreement with the experimental data of solid xenon. The method proposed in this paper can be extended to other potentials to develop practical molecular thermodynamic equations of state for solids.     

Dynamic chaos in highly excited states of an anharmonic oscillator in an external electric field

The behavior of an anharmonic quantum oscillator in a harmonic external electric field is examined. The quasienergy is calculated numerically as a function of the field parameters (intensity and frequency) for highly excited states. It depends in complex fashion on frequency and field intensity near resonance. Quasi-intersection of levels is observed in a broad range of quantum numbers. These effects are manifestations of dynamic chaos in highly excited states of an anharmonic oscillator in an external field. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 11–18, February, 1998.     

Spin force and torque in non-relativistic Dirac oscillator on a sphere

The spin force operator on a non-relativistic Dirac oscillator (in the non-relativistic limit the Dirac oscillator is a spin one-half 3D harmonic oscillator with strong spin–orbit interaction) is derived using the Heisenberg equations of motion and is seen to be formally similar to the force by the electromagnetic field on a moving charged particle. When confined to a sphere of radius R, it is shown that the Hamiltonian of this non-relativistic oscillator can be expressed as a mere kinetic energy operator with an anomalous part. As a result, the power by the spin force and torque operators in this case are seen to vanish. The spin force operator on the sphere is calculated explicitly and its torque is shown to be equal to the rate of change of the kinetic orbital angular momentum operator, again with an anomalous part. This, along with the conservation of the total angular momentum, suggests that the spin force exerts a spin-dependent torque on the kinetic orbital angular momentum operator in order to conserve total angular momentum. The presence of an anomalous spin part in the kinetic orbital angular momentum operator gives rise to an oscillatory behavior similar to the Zitterbewegung. It is suggested that the underlying physics that gives rise to the spin force and the Zitterbewegung is one and the same in NRDO and in systems that manifest spin Hall effect.     

Suppression of decoherence in a wave packet via nonlinear resonance

We propose a simple approach for suppressing decoherence of a wave packet excited in an anharmonic oscillator. We show that when a resonant external field forces the oscillator to follow the driving force, motion around the resonant trajectory inside a stable resonant island can be made almost completely immune to the environment. As an example, we study suppression of decoherence due to coupling to thermally populated rotations in vibrational wave packets in a Na2 molecule.     

势阱中粒子能级与波函数微扰计算的代数递推公式

利用超位力定理（HVT）和Hellmann-Feynman定理（HFT）,导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式,并导出由能级近似值计算定态波函数近似表达式的代数公式,给出了代数公式具体应用的几个典型一维势阱实例,此法可推广到二维势阱与三维势阱的情形。