依赖于速度的量子受迫非简谐振子的演化

运用积分算子和迭代方法求得了依赖于速度的量子受迫非简谐振子的精确解。给出了非简谐振子坐标和动量算符的时间演化。这些算符被表示成适当泛函的拉普拉斯变换及随后的逆拉普拉斯交换的形式。 关键词：     

Classical and quantum oscillators of quartic anharmonicities: second-order solution

An analytical solution up to the second order in the coupling constant λ is obtained for a classical quartic anharmonic oscillator by using Taylor series method. Our solution yields, as a special instance, the corresponding results obtained by using Laplace transform. With the help of correspondence principle, the classical solution is used to obtain the solution corresponding to a quantum quartic anharmonic oscillator. In the weak coupling regime (i.e., anharmonic constant λ⪡1), the so-called secular terms in classical and quantum solutions are tucked in (summed up) to avoid the nonconvergence. Both the classical and quantum solutions are used to obtain the frequency shifts of the quartic oscillators. It is found that these frequency shifts coincide exactly with those of the earlier results obtained by other methods. From the quantum field theoretic point of view, our solution exhibits the so-called Lamb shift. As an application of the solution for the quantum oscillator, we examine the possibility of getting squeezed states out of the input coherent light interacting with a nonlinear medium of inversion symmetry.     

Exact solutions of the Klein—Gordon equation with ring-shaped oscillator potential by using the Laplace integral transform

We present exact solutions for the Klein-Gordon equation with a ring-shaped oscillator potential. The energy eigenvalues and the normalized wave functions are obtained for a particle in the presence of non-central oscillator potential. The angular functions are expressed in terms of the hypergeometric functions. The radial eigenfunctions have been obtained by using the Laplace integral transform. By means of the Laplace transform method, which is efficient and simple, the radial Klein-Gordon equation is reduced to a first-order differential equation.     

Unified Method for Dynamical Groups of Some Anharmonic Potentials

Realizations of the creation and annihilation operators for some important anharmonic potentials, such as the Morse potential, the modified Pöschl–Teller potential (MPT), the pseudoharmonic oscillator, and infinitely deep square-well potential, are presented by a factorization method. It is shown that the operators for the Morse potential and the MPT potential satisfy the commutation relations of an SU(2) algebra, but those of the pseudoharmonic oscillator and the infinitely deep square-well potential constitute an SU(1, 1) algebra. The matrix elements of some related operators are analytically obtained. The harmonic limits of the SU(2) operators for the Morse and MPT potentials are studied as the Weyl algebra.     

A model for a stochastic multiphoton absorption process of an anharmonic oscillator

A model of multiphoton absorption of i.r. laser radiation by an anharmonic, molecular, vibrational mode is discussed. The multiphoton absorption is described as a stochastic process for which the reservoir, including rotation and other vibrational modes, modulates the frequency of the active oscillator stochastically. The treatment of the nonlinear oscillator is based on quantization of a classical, driven anharmonic oscillator.     

Splitting of Spectra in Anharmonic Oscillators Described by Kratzer Potential Function

A perturbation theory model that describes splitting of the spectra in highly symmetrical molecular species in electrostatic field is proposed. An anahrmonic model of a two-dimensional oscillator having Kratzer potential energy functionis used to model the molecular species and to represent the unperturbed system. A selection rule for the radial quantum number of the oscillator is derived. The eigenfunctions of a two-dimensional anharmonic oscillator in cylindrical coordinates are used for the matrix elements representing the probability for energy transitions in dipole approximation to be calculated. Several forms of perturbation operators are proposed to model the interactionbetween the polyatomic molecular species and an electrostatic field. It is found that the degeneracy is removed in the presence of the electric field and spectral splitting occurs. Anharmonic approximation for the unperturbed system is more accurate and reliable representation of a real polyatomic molecular species.     

Energy features of an adiabatically loaded anharmonic oscillator

An excited anharmonic oscillator is considered under conditions of adiabatic (i.e., slow, as compared to the oscillation period) loading with an external force tending to a constant value at long times. The energy characteristics of the adiabatically loaded anharmonic oscillator, such as the instantaneous energy of the oscillator, the maximum kinetic (oscillation) energy, and the kinetic and potential energies averaged over the period, are analytically calculated as a function of the steady-state force. The analytical results are confirmed by the data of numerical calculations. It is established that the external force gives rise to a redistribution of the average kinetic and potential components of the initial energy of the anharmonic oscillator and that the transferred energy portions at a small external force considerably exceed the average work done by the external force.     

On the anharmonic oscillator in quantum mechanics and in field theory

A modified Rayleigh-Schrödinger perturbation method is used to derive explicit expansions for the eigenvalues and eigensolutions of the anharmonic oscillator. We then point out the dual relationship between the anharmonic oscillator and the Schrödinger equation for a Yukawa potential. Finally we consider an application of the method to a field-theoretic Hamiltonian, since the anharmonic oscillator plays a dominant role in many field-theoretic models.     

有限维q非谐振子广义相干态振幅N次方压缩

利用Zhang等人提出的单模辐射场的振幅N次方压缩理论,研究了有限维希尔伯特空间中单参数q变形非简谐振子广义相干态的振幅N次方压缩效应.结果发现:该量子光场的确存在场的振幅N次方压缩效应,其压缩条件分别与相位角Φ、维度参数s、压缩幂次N、平均光子数的方根r和变形参数q相关;这一结果与无限维希尔伯特空间单参数q变形广义相干态的情形截然不同.     

The ground state energy per particle for infinite particle quantum systems

We give a simple proof that the ground state energy per particle for several interacting particle systems is monotone and bounded as the number of particles increases. Some of the systems for which the proof holds are anharmonic oscillator approximations to || _d ^/4 quantum fields, many body Schrödinger operators with nearest and next to nearest neighbor couplings, and systems whose energy is given by operators which are not restricted to being differential operators.Research partially supported by the National Science Foundation under grant No. MCS-77-03568     

On the p-adic summability of the anharmonic oscillator

Some remarkable arithmetic properties of the anharmonic oscillator are considered. The concept of the p-adic summability is discussed. For the ground-state energy of the anharmonic oscillator the p-adic summability is shown.     

用算符分解方法计算非同调谐振子几何相位及其推广

将几何相位理论应用于非同调谐振子(isotonic oscillator缩写为IO)这类量子系统,运用算符分解方法计算了系统在二态体系的Aharonov-Anandan相位,推广至三态及多态体系,并讨论了AA相位更普遍的计算公式和变化规律.     

三维各向同性谐振子波函数的递推关系(英文)

本文应用拉普拉斯变换得到了三维各向同性谐振子波函数边界的精确解,同时,利用同种方法还得到了利用产生算符和湮灭算符表达的该波函数的递推关系.     

Kinetic theory of a novel carm oscillator with axisymmetrical quasi—Optical cavity of oblique rotation at arbitrary angle

A novel CARM oscillator with axisymmetrical quasi—optical cavity of oblique rotation at arbitrary angle (AQCORAA) is studied using the kinetic theory. By means of the method of local field expansion in the coordinates of the guiding centre and the Laplace transform, the beam—wave interaction power, starting current density and frequency shift are derived. The formulas presented here may have some potential applications for more further studies of the CARM oscillator with AQCORAA.This work is supported by National Science Fund of China, and by Doctoral Fund of Education Committee of China.     

A new fractional derivative involving the normalized sinc function without singular kernel

In this paper, a new fractional derivative involving the normalized sinc function without singular kernel is proposed. The Laplace transform is used to find the analytical solution of the anomalous heat-diffusion problems. The comparative results between classical and fractional-order operators are presented. The results are significant in the analysis of one-dimensional anomalous heat-transfer problems.     

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：     

A New Quasi-Exactly Solvable Problem and Its Connection with an AnharmonicOscillator

The two-dimensional hydrogen with a linear potential in a magnetic field is solved by two different methods. Furthermore the connection between the model and an anharmonic oscillator is investigated by methods of KS transformation.     

Thermal 1/ω fluctuations of a quantum oscillator under a parametric effect of a random field

It is shown that the effect of a time-correlated Gaussian random field of sufficiently high intensity on the elasticity coefficient of a quantum oscillator manifests itself in the generation of thermal fluctuations with a 1/ω spectrum in the oscillator. It is also shown that, in any physical system described by the equation of an anharmonic oscillator, fluctuations with a 1/ω spectrum arise at an above-critical temperature.