阻尼谐振子到简谐振子的变换

在牛顿力学、拉格朗日力学和哈密顿力学3种形式中,利用变量变换将阻尼谐振子变换成简谐振子.     

Non-Heisenberg states of the harmonic oscillator

The effects of the vacuum electromagnetic fluctuations and the radiation reaction fields on the time development of a simple microscopic system are identified using a new mathematical method. This is done by studying a charged mechanical oscillator (frequency ₀)within the realm of stochastic electrodynamics, where the vacuum plays the role of an energy reservoir. According to our approach, which may be regarded as a simple mathematical exercise, we show how the oscillator Liouville equation is transformed into a Schrödinger-like stochastic equation with a free parameter h with dimensions of action. The role of the physical Planck's constant h is introduced only through the zero-point vacuum electromagnetic fields. The perturbative and the exact solutions of the stochastic Schrödinger-like equation are presented for h>0. The exact solutions for which h<h are called sub-Heisenberg states. These nonperturbative solutions appear in the form of Gaussian, non-Heisenberg states for which the initial classical uncertainty relation takes the form (x ²)(p) ² =(h/2) ²,which includes the limit of zero indeterminacy (h 0). We show how the radiation reaction and the vacuum fields govern the evolution of these non-Heisenberg states in phase space, guaranteeing their decay to the stationary state with average energy h ₀ /2 and (x) ² (p) ² =h ² /4 at zero temperature. Environmental and thermal effects-are briefly discussed and the connection with similar works within the realm of quantum electrodynamics is also presented. We suggest some other applications of the classical non-Heisenberg states introduced in this paper and we also indicate experiments which might give concrete evidence of these states.     

A no-go result for the quantum damped harmonic oscillator

In this letter we show that it is not possible to set up a canonical quantization for the damped harmonic oscillator using the Bateman Lagrangian. In particular, we prove that no square integrable vacuum exists for the natural ladder operators of the system, and that the only vacua can be found as distributions. This implies that the procedure proposed by some authors is only formally correct, and requires a much deeper analysis to be made rigorous.     

A time-discrete harmonic oscillator model of human car-following

A time-discrete stochastic harmonic oscillator is presented as a model of human car-following behaviour. This describes especially the non-continuous control of a human driver – acceleration changes from time to time at so called action-points and is kept constant in between. Analytical results can be derived which allow to classify the different types of motion possible within this approach. These results show that with weaker control by the human, unstable behaviour of the oscillator becomes more likely. This is in line with common understanding about the causes of accidents. Finally, since even the stochastic behaviour of this model is in parts analytically tractable, the width of the speed-difference and distance fluctuations can be expressed as function of the model’s parameter. This allows a fresh view on empirical car-following data and the identification of parameters from real data in the context of the theory presented here.     

Hemispheroidal quantum harmonic oscillator

A deformed single-particle shell model is derived for a hemispheroidal potential well. Only the negative parity states of the Z(z) component of the wave function are allowed, so new magic numbers are obtained. The influence of a term proportional to l² in the Hamiltonian is investigated. The maximum degeneracy is reached at a superdeformed hemispheroidal prolate shape whose magic numbers are identical with those obtained at the spherical shape of the spheroidal harmonic oscillator. This remarkable property suggests an increased stability of such a distorted shape of deposited clusters when the planar surface remains opaque.     

Multiphoton transitions for the harmonic oscillator

2- and 3-photon electric dipole transitions for the harmonic oscillator are shown to have zero intensity. This is extended to the general cases of Δv?=?n and Δv?=?n???2 for n-photon transitions. Although not proved in general, it seems likely that all multiphoton transition moments vanish. This suggests that electric dipole multiphoton transition intensity is due to anharmonicity.     

Quantum decoherence of the damped harmonic oscillator

In the framework of the Lindblad theory for open quantum systems, we determine the degree of quantum decoherence of a harmonic oscillator interacting with a thermal bath. It is found that the system manifests a quantum decoherence which is more and more significant in time. We also calculate the decoherence time and show that it has the same scale as the time after which thermal fluctuations become comparable with quantum fluctuations. The text was submitted by the author in English.     

Background independent duals of the harmonic oscillator

We show that a class of topological field theories are quantum duals of the harmonic oscillator. This is demonstrated by establishing a correspondence between the creation and annihilation operators and nonlocal gauge invariant observables of the topological field theory. The example is used to discuss some issues concerning background independence and the relation of vacuum energy to the problem of time in quantum gravity.     

Quadrupole perturbation of the harmonic oscillator potential

TheSU(3) symmetry of the harmonic oscillator potential is used to calculate exactly the second order energy correction, caused by quadrupole deformations. This can be done in a purely algebraic manner, using the Dalgarno-Schwartz formulation of perturbation theory. Some connections of this method with group theory are discussed and an extension to more general situations is proposed.     

Complete solution of the cranked harmonic oscillator

A complete solution of the cranked harmonic oscillator is given in a model in which the energy is minimized with constant volume and constant angular momentum. Two extreme regimes emerge for the configurationsΣ ₁,Σ ₂,Σ ₃: at low 〈L ₁〉 the nucleus tends to become more symmetric around its rotation axis, at high 〈L ₁〉 its elongation is increased. The oblate regime marks the separation between these two. The condition of stability of the oblate regime is discussed.     

Quantum theory of the damped harmonic oscillator

A phenomenological stochastic modelling of the process of thermal and quantal fluctuations of a damped harmonic oscillator is presented. The divergence of the momentum dispersion associated with the Markovian limit is removed by a Drude regularization. The variances of position and momentum are evaluated in closed form at arbitrary temperature and for arbitrary damping. Properties of real and imaginary time correlation functions are discussed, and a spectral decomposition of the equilibrium density matrix is given.     

q-形变的非简谐振子广义相干叠加态的高阶压缩效应

研究了q-形变的非简谐振子的广义相干态的高阶压缩效应,给出了出现高阶压缩现象的条件.     

Coupled N-dimensional harmonic oscillator systems

A general decoupling procedure is described for systems of N interacting harmonic oscillators coupled by some bilinear perturbing potentials, and explicit conditions are derived which must be fulfilled in order to obtain physical bound states. Illustrative examples include a general treatment of the 2-oscillator system, some particular extensions to systems containing 3 oscillators, and a class of N-oscillator models.     

A concise quantum mechanical treatment of the forced damped harmonic oscillator

By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator \(\hat X\) in x representation. The evolution of \(\left\langle {\hat x} \right\rangle \) is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.     

On the eigenvalues of the rotating harmonic oscillator

The eigenvalues of the radial Schrödinger equation of a rotating harmonic oscillator depending on a parameter α > 0 are shown to be independent of α and equal to integers.