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1.
We introduce the Schrödinger correlator as the holistic characteristic of two types of fluctuation correlations in quantum dynamics and in statistical thermodynamics. We are the first to derive it using methods of thermofield dynamics for the coordinate-momentum variables of a quantum oscillator in a thermostat. We show that the obtained value ensures that the Schrödinger uncertainty relation becomes an equality at all temperatures. We find that the thermal equilibrium for the quantum oscillator has the sense of the thermal correlated coherent state and can be adequately described by a wave function with temperature-dependent amplitude and phase.  相似文献   

2.
We consider a new completely integrable case of the time-dependent Schrödinger equation in ®n with variable coefficients for a modified oscillator that is dual (with respect to time reversal) to a model of the quantum oscillator. We find a second pair of dual Hamiltonians in the momentum representation. The examples considered show that in mathematical physics and quantum mechanics, a change in the time direction may require a total change of the system dynamics to return the system to its original quantum state. We obtain particular solutions of the corresponding nonlinear Schrödinger equations. We also consider a Hamiltonian structure of the classical integrable problem and its quantization.  相似文献   

3.
In this paper, we construct an exact solution of the stochastic Schrodinger equation for a quantum oscillator with possible dissipation of energy taken into account. Using the explicit form of the solution, we calculate estimates for the characteristic damping time of free damped oscillations. In the case of forced oscillations, we obtain formulas for the Q-factor of the system and for the variance of the coordinate and momentum of a quantum oscillator with dissipation. We obtain the quantum analog of the classical diffusion equation and explicitly show that the equations of motion for the mean value of the momentum operator following from the solution of the stochastic Schrodinger equation play the role of the quantum Langevin equation describing Brownian motion under the action of a stochastic force.  相似文献   

4.
Novikova  E. M. 《Mathematical Notes》2019,106(5-6):940-956
Mathematical Notes - The algebra of symmetries of a quantum three-frequency hyperbolic resonance oscillator is studied. It is shown that this algebra is determined by a finite set of generators...  相似文献   

5.
Summary We prove an existence, uniqueness and unitarity theorem for quantum stochastic differential equations with unbounded coefficients which satisfy an analyticity condition on a common dense invariant domain. This result, applied to the quantum harmonic oscillator, gives a rigorous meaning to a large class of stochastic differential equations that have been considered formally in quantum probability.  相似文献   

6.
In this work it is shown that the intrinsic phenomenon (the quantization of the energy) that appears in the first and simple systems studied initially by the quantum theory as the harmonic oscillator and the movement of a charged particle under the Coulomb force, can be obtained from the study of dissipative systems. In others words, we show that this phenomenon of the quantization of the energy of a particle which moves as an harmonic oscillator and which loses and wins energy can be obtained via a classical system of equations. The same also applies to the phenomena of the quantization of the energy of a charged particle which moves under the Coulomb force and which loses and wins energy.  相似文献   

7.
In this work it is shown that the intrinsic phenomenon (the quantization of the energy) that appears in the first and simple systems studied initially by the quantum theory as the harmonic oscillator and the movement of a charged particle under the Coulomb force, can be obtained from the study of dissipative systems. In others words, we show that this phenomenon of the quantization of the energy of a particle which moves as an harmonic oscillator and which loses and wins energy can be obtained via a classical system of equations. The same also applies to the phenomena of the quantization of the energy of a charged particle which moves under the Coulomb force and which loses and wins energy.  相似文献   

8.
9.
We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(?1) and 1:(?2). To cite this article: N.N. Nekhoroshev et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988.  相似文献   

10.
We consider an extension of the Feynman path integral to the quantum mechanics of noncommuting spatial coordinates and formulate the corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians). The basis of our approach is that a quantum mechanical system with a noncommutative configuration space can be regarded as another effective system with commuting spatial coordinates. Because the path integral for quadratic Lagrangians is exactly solvable and a general formula for the probability amplitude exists, we restrict our research to this class of Lagrangians. We find a general relation between quadratic Lagrangians in their commutative and noncommutative regimes and present the corresponding noncommutative path integral. This method is illustrated with two quantum mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.  相似文献   

11.
In this work an observation concerning a positivity property of the quantum anharmonic oscillator is made. This positivity property is suggested by the BMV conjecture.  相似文献   

12.
Novikova  E. M. 《Mathematical Notes》2021,109(5-6):777-793
Mathematical Notes - For the perturbed Hamiltonian of a multifrequency resonance harmonic oscillator, a new approach to calculating the coefficients in the procedure of quantum averaging is...  相似文献   

13.
The behavior of a quantum oscillator in an infinite-particle system is studied for the case of linear interaction. The relation between the spectrum of the dynamic matrix of a complete system and oscillator damping is established. The dependence of the spectrum on the parameters of interaction is determined.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1266–1273, September, 1993.  相似文献   

14.
The concept of the fractional Fourier transform is framed withinthe context of quantum evolution operators. This point of viewyields an extension of the above concept and greatly simplifiesthe underlying operational algebra. It is also proved that amultidimensional extension can be performed by using a biorthogonalmultiindex harmonic oscillator basis. It is finally shown thatmost of the proposed physical interpretations of the fractionalFourier transform are just trivial consequences of the analysisdeveloped in this paper.  相似文献   

15.
The classical and quantum formalism for a p-adic and adelic harmonic oscillator with a time-dependent frequency is developed, and general formulas are obtained for the main theoretical quantities. In particular, the p-adic propagator is calculated, and the existence of a simple vacuum state as well as adelic quantum dynamics is shown. A spatial discreteness and a p-adic quantum mechanical phase are noted. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 239–248, August, 2000.  相似文献   

16.
《Quaestiones Mathematicae》2013,36(4):387-393
ABSTRACT

In this article we apply the geometric quantization method of Rund to the case of the three-dimensional harmonic oscillator and show that the eigenvalue spectrum thus obtained coincides precisely with the energy eigenvalue spectrum that is prescribed by standard quantum mechanics.  相似文献   

17.
In this paper, the deformation of the ordinary quantum mechanics is formulated based on the idea of conformable fractional calculus. Some properties of fractional calculus and fractional elementary functions are investigated. The fractional wave equation in 1 + 1 dimension and fractional version of the Lorentz transformation are discussed. Finally, the fractional quantum mechanics is formulated; infinite potential well problem, density of states for the ideal gas, and quantum harmonic oscillator problem are discussed.  相似文献   

18.
Adelic quantum mechanics is formulated. The corresponding model of the harmonic oscillator is considered. The adelic harmonic oscillator exhibits many interesting features. One of them is a softening of the uncertainty relation.This work was partially supported by the Russian Foundation for Fundamental Research, Grant No. 93-011-147.Institute of Physics, P.O. Box 57, 11001 Belgrade, Yugoslavia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 3, pp. 349–359, December, 1994.  相似文献   

19.
Karasev  M. V.  Novikova  E. M. 《Mathematical Notes》2018,104(5-6):833-847
Mathematical Notes - For the three-frequency quantum resonance oscillator, the reducible case, where the frequencies are integer and at least one pair of frequencies has a nontrivial common...  相似文献   

20.
The optimal control for cooling a quantum harmonic oscillator by controlling its frequency is considered. It is shown that this singular problem may be transformed with the proper choice of coordinates to an equivalent problem which is no longer singular. The coordinates used are sufficiently simple that a graphical solution is possible and eliminates the need to use a Weierstrass-like approach to show optimality. The optimal control of this problem is of significance in connection with cooling physical systems to low temperatures. It is also mathematically significant in showing the power and limitations of coordinate transformations for attacking apparently singular problems.  相似文献   

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