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.     

Statistical thermodynamics of an anharmonic oscillator

In the present study we investigate the statistical thermodynamics of the anharmonic oscillator, whose energies are characterized by the potential 1/2x ²+x ⁴. Employing the energies recently obtained by Hioe and Montroll, we compute the partition function and the thermodynamic quantities for the anharmonic and quartic oscillators. Low- and high-temperature formulas are presented for the thermodynamic quantities of the oscillators.     

Radiation damping in an anharmonic oscillator

The effect of radiation damping in an anharmonic oscillator has been calculated using the technique of Kryloff and Bogolituboff. It is found that the intensity distribution of the emitted spectral line is asymmetric about its intensity maximum. The index of asymmetry agree with the experimental data on x-rayK _{a 1,2}lines.     

Energy of the fundamental level of an anharmonic oscillator in the strong-coupling limit

A nonperturbative method of calculating functional integrals is developed. The effectiveness of its application is illustrated for the example of calculating the energy of the fundamental level of an anharmonic oscillator.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 64–69, July, 1990.It remains to thank A. A. Afonin, V. N. Kapshai, V. N. Starikov, and V. G. Teplyakov for interest in the work and useful discussions of the results.     

Quasiclassical trajectory-coherent states of an anharmonic oscillator

Based on the complex WKB—Maslov method, asymptotic solutions accurate within O(h²) are constructed for a class of wave packets — quasiclassical trajectory-coherent states for a quantum anharmonic oscillator with a potential V(x)=x²+x⁴. We obtain the behavior of the packet width and the deviations of the approximate solution as a function of time. The actual parameter of the quasiclassical expansion is shown to be h^1/2. We find the regions of variation of the potential parameters and of the initial data for which is small, while the breakdown time of the given accuracy and the packet spreading time are comparable to the time of a single particle revolution.Moscow Institute of Electric Machine Construction. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 21–27, September, 1993.     

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)     

Energy levels of a two-dimensional anharmonic oscillator: Hill determinant approach

The energy levels of the Schrödinger equation for the potentialx ²+y ²+λ[a _xxx⁴+2a _xyx² y ²+a _yyy⁴] are calculated using Hill determinant approach for several eigenstates and over a wide range of values of perturbation parameters. The obtained numerical results agree with those previously reported by other methods.     

Diagonal vibrational matrix elements for an anharmonic oscillator

General expressions for the diagonal rotationless matrix elements of any power of the internuclear displacement from equilibrium are obtained for ¹Σ-state diatomic molecule using an iterative relation. These expressions, including contributions arising from terms of the ninth power in the Dunham potential, are obtained in analytical form. As an application, sixth- and seventh-order contributions to the matrix element 〈0?x¹?〉 are calculated for the first eight powers of the displacement.     

On perturbation theory for an asymmetric anharmonic oscillator

A modified perturbation theory is formulated for an asymmetric anharmonic oscillator based on a choice of the main approximation constructed by analogy with the self-consistent field model. This perturbation theory has a much wider range of application in comparison with the standard approach and considers a finite number of energy levels in the potential well already in the main approximation. This approach is used for the construction of a two-atomic model of a quantum crystal. A quantum analog of the Lindeman criterion is obtained. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 30–40, January, 2009.     

Quantum and classical chirps in an anharmonic oscillator

We measure the state dynamics of a tunable anharmonic quantum system, the Josephson phase circuit, under the excitation of a frequency-chirped drive. At small anharmonicity, the state evolves like a wave packet-a characteristic response in classical oscillators; in this regime, we report exponentially enhanced lifetimes of highly excited states, held by the drive. At large anharmonicity, we observe sharp steps, corresponding to the excitation of discrete energy levels. The continuous transition between the two regimes is mapped by measuring the threshold of these two effects.     

Dynamics of an anharmonic oscillator with a periodic perturbation

We study the features of the stochastic dynamics of a Hamiltonian system with the potential x ²ⁿ subjected to an external monochromatic perturbation. Three regimes of stochastic diffusion, which differ in the value of the amplitude of the external perturbation, are detected. We demonstrate the possibility of chaotic regimes manifesting themselves in pendulum vibrations of the well of a water-moderated and-cooled nuclear power reactor as an application of the model being investigated. Finally, we propose a method of simple proportional control, which makes it possible to control the chaotic vibrations of the anharmonic oscillator. Zh. éksp. Teor. Fiz. 115, 361–377 (January 1999)     

A supersymmetric anharmonic oscillator

We consider the supersymmetric model obtained from the Wess-Zumino model with interaction, with the help of a dimensional reduction. The solution is found of the corresponding classical problem with a precision up to terms of the third order of smallness. A quantum model of such a system is constructed. Its spectrum is found in the first and second order of perturbation theory. The supersymmetric finite-dimensional model under consideration proves to be extremely close to the usual anharmonic oscillator.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 28–32, May, 1990.     

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.     

Classical chaos and quantum level density of an anharmonic oscillator

The Liapunov exponents of two-dimension anharmonic oscillator systems are studied through numerical calculations. The result shows that the systems consist of regular and irregular regions in phase space in the classical limit. The corresponding quantum systems are investigated. The distribitionP(s) of spacings between adjacent energy levels indicates a corresponding transition from Poisson-like distribution to Wigner-like distribution.P(s) is dependent on the total irregular fraction of phase space.     

Energy transfer to an anharmonic diatomic system

We model an anharmonic diatomic molecule using deformed creation and annihilation operators such that the energy spectrum generated by a Hamiltonian of the harmonic oscillator's form written in terms of deformed operators is similar to that of a Morse potential. We construct an approximate time evolution operator and evaluate transition probabilities which are compared with those obtained by an expansion in a basis of Morse eigenfunctions. The algebraic results compare favorably with the numerical results.     

A perturbation theory for identifying quasi-coupled states of an anharmonic oscillator

A perturbation theory designed for calculating quasi-coupled or coupled state energies and which is valid at any value of the coupling parameter is constructed in an example of an anharmonic oscillator with quartic nonlinearity.     

Pade approximants and the anharmonic oscillator

The diagonal Padé approximants of the perturbation series for the eigenvalues of the anharmonic oscillator (a βκ¹ perturbation of p² + κ²) converge to the eigenvalues.     

Spectral singularities of the quartic anharmonic oscillator

The singularities of the eigenvalues of the quartic anharmonic oscillator hamiltonian are studied numerically as a function of the quartic coupling parameter. Their location is found to be consistent with previous WKB calculations but the identification of which levels are singular is found to differ.