Structures symplectiques, structures complexes, spineurs symplectiques et transformation de Fourier

Planck's constant is very useful in the development of the theory of symplectic Clifford algebras introduced by the author in 1977 [1,a], and to solve many connected problems for example the Poisson Lie algebra deformations [1,c]. In this paper we give a precise link between a complex structure J and the Fourier transform which is nothing but the natural left action of the covering J? of J in a symplectic convenient spinor space (modulo a constant factor).Thus Fourier transform becomes a geometric transformation separated from integration technics, good peculiarity for global problems. We explain nice algebraic properties of the Fourier transform taking them in the symplectic context with adapted metric in any signature. Some applications are given: Hermite's functions, Plancherel-Parseval's theorem, covariance problemes … . Our approach is particularly convenient for explain results in Maslov's theory [1,b] and the difficulties in defining a global Fourier transform over a symplectic manifold.     

Evolution of Wigner's phase-space density under a nonintegrable quantum map

A nonintegrable area-preserving map for a system with one freedom is quantized, and the evolution of Wigner's function W(q,p) illustrated by contour plots of W in the paase plane. In the classical limit, propagation is governed by Liouville's equation and the contours of W rapidly develop an intricate structure of whorls and tendrils. When Planck's constant ? is not zero, the quantum map smooths out classical detail in phase-space areas smaller than ?. The quantum-mechanical distributions spread more slowly than their classical counterparts.     

Massless Spin–Zero Particle and the Classical Action via Hamilton–Jacobi Equation in G?del Universe

In this letter we investigate the separability of the Klein–Gordon and Hamilton–Jacobi equation in G?del universe. We show that the Klein–Gordon eigen modes are quantized and the complete spectrum of the particle’s energy is a mixture of an azimuthal quantum number, m and a principal quantum number, n and a continuous wave number k. We also show that the Hamilton–Jacobi equation gives a closed function for classical action. These results may be used to calculate the Casimir vacuum energy in G?del universe.     

Quantum coherent oscillations between two coupled bose-einstein condensates

The theoretical investigation of quantum coherent atomic oscillations between two coupled Bose-Einstein condensates(BECs) is studied. We apply the inseparable wave function of time-space to describe two trapped BECs in a double-well magnetic trap. According to Thomas-Fermi approximation, dynamical equations of the interwell phase difference and population imbalance are obtained. Using numerical method, coherent atomic tunneling and macroscopic quantum self-trapping(MQST) effect are investigated.     

两耦合玻色-爱因斯坦凝聚体间的量子相干振荡

理论上考察了两耦合玻色-爱因斯坦凝聚体间的相干原子振荡,我们用时空不能完全分离的波函数去描述囚禁在双磁阱中的玻色-爱因斯坦凝聚体,根据托马斯-费米近似,得到两凝聚体的相位差和布局数随时间的演化方程,应用数值计算的方法,考察了相干原子遂穿和宏观量子自囚禁效应.这些研究结果和采用双模时空分离波函数近似法得到的结果进行了比较.     

A classical theory of superparamagnetic relaxation

A Fokker-Planck equation for the relaxation of a classical ferromagnetic particle coupled to a classical heat bath is derived from the Nakajima-Zwanzig equation. The equation of motion for the mean magnetization of an ensemble of particles is found to be closed only under special circumstances. In the strong motional narrowing limit the equation of motion reduces to the Bloch equations in the limit MH ? k_BT, i.e. for small particles, and to the Landau-Lifshitz equation in the opposite limit. For the motional narrowing region in toto the particular case of uniaxial anisotropy is analysed, giving an equation of motion which for large particles reduces to a modified Landau-Lifshitz equation with g-shift and a reduced damping constant. This equation cannot be meaningfully identified with the Gilbert equation.Approximate expressions for superparamagnetic relaxation rates by Kramers' method are obtained for the case of (i) triaxial (i.e. orthorhombic) and (ii) cubic (K +ve and ?ve) anisotropy, assuming large energy barriers. The results supplement Brown's expression for uniaxial anisotropy and show a more complicated dependence on the Landau-Lifshitz parameter λ than the linear dependence found by Brown. For small λ the rates tend to constant values compatible with the transition.     

Semiclassical quantization of the nonlinear Schrödinger equation

Using the functional integral technique of Dashen, Hasslacher, and Neveu, we perform a semiclassical quantization of the nonlinear Schrödinger equation, which reproduces McGuire's exact result for the energy levels of the theory's bound states. We show that the stability angle formalism leads to the one-loop normal ordering and self-energy renormalization expected from perturbation theory and demonstrate that taking into account center-of-mass motion gives the correct nonrelativistic energymomentum relation. We interpret the classical solution in the context of the quantum theory, relating it to the matrix element of the field operator between adjacent bound states in the limit of large quantum numbers. Finally, we quantize the NLSE as a theory of N component fermion fields and show that the semiclassical method yields the exact energy levels and correct degeneracies.     

Families of bose rays in quantum optics

Having known classical wave optics and wave mechanics, can we reverse Schrödinger's path and extend the concept of families of rays of light to provide a new exact rendering of quantum optics including the Bose nature of photons? This question is answered in the affirmative, and the implications of the Bose symmetry for certain nonlocal correlations of the many-ray distribution functions are worked out. The similarities and the differences between classical and quantum wave optics are brought out. The ray-ray Bose correlation is analyzed. The generating functional for the many-ray distribution functions is formulated; and the notion of paraxial illumination for quantum optics is made precise.     

Effects of magnetic field and spin-orbit interaction on energy levels in 1D quantum wire: analytical solution

In this paper, we study the effects of a perpendicular magnetic field and spin-orbit interaction (SOI) on energy levels of a quasi-one-dimensional quantum wire. Here, we solve analytically the Schr?dinger equation to obtain energy levels and wave functions. The results show that the splitting at k?=?0 is not constant and it depends on the subband. This variable splitting is due to a symmetry breaking effect when magnetic field and SOI are simultaneously present.     

Nonlinearity of Quantum Mechanics and Solution of the Problem of Wave Function Collapse

The problem of the wave function collapse(a problem of measurement in quantum mechanics) is considered.It is shown that it can be solved based on quantum mechanics and does not require any additional assumptions or new theories. The particle creation and annihilation processes, which are described based on quantum field theory, play a key role in the measurement processes. Superposition principle is not valid for the system of equations of quantum field theory for particles and fields, because this system is a non-linear. As a result of the creation(annihilation) of a particle,an additional uncertainty arises, which "smears" the interference pattern. The imposition of such a large number of uncertainties in the repetitive measurements leads to the classical behavior of particles. The decoherence theory also implies the creation and annihilation of particles, and this processes are the consequence of non-linearity of quantum mechanics. In this case, the term "collapse of the wave function" becomes a consequence of the other statements of quantum mechanics instead of a separate postulate of quantum mechanics.     

Exact uncertainty approach in quantum mechanics and quantum gravity

The assumption that an ensemble of classical particles is subject to nonclassical momentum fluctuations, with the fluctuation uncertainty fully determined by the position uncertainty, has been shown to lead from the classical equations of motion to the Schrödinger equation. This ‘exact uncertainty’ approach may be generalised to ensembles of gravitational fields, where nonclassical fluctuations are added to the field momentum densities, of a magnitude determined by the uncertainty in the metric tensor components. In this way one obtains the Wheeler-DeWitt equation of quantum gravity, with the added bonus of a uniquely specified operator ordering. No a priori assumptions are required concerning the existence of wave functions, Hilbert spaces, Planck's constant, linear operators, etc. Thus this approach has greater transparency than the usual canonical approach, particularly in regard to the connections between quantum and classical ensembles. Conceptual foundations and advantages are emphasised.     

Self-Trapping of Acoustic Polaron in One Dimension

The ground-state energy and effective mass of an acoustic polaron in one dimension are calculated by using an electron-longitudinal-acoustic-phonon interaction Hamiltonian derived here. The self-trapping of the acoustic polaron is discussed. It is found that the critical coupling constant shifts toward weaker electron-phonon interaction with the increasing cutoff wave vector and the products of the critical coupling constant by the cutoff wave vector tend to a certain value. The self-trapping of acoustic polarons in one dimension is easier to be realized than that in three- and two-dimensional systems. The self-trapping transition of acoustic polarons is expected to be observed in the one dimensional systems of alkali halides and wide-band-gap semiconductors.     

TCP, quantum gravity,the cosmological constant and all that...

We study cosmology from the point of view of quantum gravity. Some light is thrown on the nature of time, and it is suggested that the cosmological arrow of time is generated by a spontaneous breakdown of TCP. Conventional cosmological models in which quantum fields interact with a time-dependent gravitational field are shown to describe an approximation to the quantum gravitational wave function which is valid in the long-wavelength limit. Two problems with initial conditions are resolved in models in which a negative bare cosmological constant is cancelled by the classical excitation of a Bose field η with a very flat potential. These models can also give a natural explanation for the observed value of the cosmological constant.     

一维双组分玻色-爱因斯坦凝聚体系的量子隧穿特性

利用双模近似方法研究了一维双组分玻色-爱因斯坦凝聚体(Bose-Einstein condensates,BECs)的量子隧穿特性.从描述三维双组分BECs系统的Gross-Pitaevskii方程(GPE)出发,得到了描述一维体系的GP方程.把体系波函数写成原子数和相位指数的乘积,得到描述体系隧穿特性的费曼方程.数值求解费曼方程,研究了原子之间相互作用(双组分BECs体系原子之间的相互作用包括组分内部原子之间的相互作用和不同组分原子之间的相互作用)对隧穿特性的影响.结果显示,当原子之间的相互作用较弱时,体系发生量子隧穿现象,表现为原子数在平衡位置附近作周期振荡;随着原子之间相互作用增强,体系经历一个临界状态,进入自俘获状态,即由于原子之间相互作用的存在,在对称双势阱中演化的BECs可以呈现出原子数高度的不对称分布,好像绝大数原子被其中一个势阱俘获.从隧穿到自俘获原子之间的相互作用存在一个临界值,从而体系的能量也对应一个临界值,根据体系的哈密顿函数,就能求出相互作用临界值的表达式.     

Magnetic properties of a nearly classical one-component plasma in three or two dimensions

The magnetic properties of the one-component plasma, in three and two dimensions, are studied in the nearly classical case. Since magnetism is an essentially quantum effect, Planck's constant ? can be used as a small parameter. A generalized Wigner-Kirkwood expansion in powers of ?² is derived. This expansion is used for computing the induced magnetization. The displacement of the liquid-solid phase transition of the model, when a magnetic field is applied, is discussed. The model is applicable to electrons deposited at the surface of liquid helium.     

Classical and Quantum Evolution of the Bianchi Type I Model

The classical and quantum evolution of an anisotropic cosmological Bianchi type I model is considered. In the classical case, the influence of the minimally coupled scalar field is taken into account. Thus the system of two equations is obtained, which are explored at the inflationary and scalaron stages. The quantum problem in view of the positive cosmological constant is considered. The principal moment of the account of an anisotropy is the occurrence of the potential barrier unbounded in zero and at infinity. Though the greatest value of the potential is less than zero and the total energy of the Universe E=0, there is an important opportunity for above-barrier reflection of the wave function of the Universe. After reflection the wave function describes the expanding Universe promptly losing anisotropy and transferring into the Friedmann Universe.     

The dynamics of triple-well trapped Bose-Einstein condensates with atoms feeding and loss effects

In this paper, we consider the macroscopic quantum tunnelling and self-trapping phenomena of Bose-Einstein condensates （BECs） with three-body recombination losses and atoms feeding from thermal cloud in triple-well potential. Using the three-mode approximation, three coupled Gross-Pitaevskii equations （GPEs）, which describe the dynamics of the system, are obtained. The corresponding numerical results reveal some interesting characteristics of BECs for different scattering lengths. The self-trapping and quantum tunnelling both are found in zero-phase and ：r-phase modes. Furthermore, we observe the quantum beating phenomenon and the resonance character during the self-trapping and quantum tunnelling. It is also shown that the initial phase has a significant effect on the dynamics of the system.     

Quantum effects in radiation on short bunches

The radiation caused by particles of one bunch in the collective electromagnetic field of the short oncoming bunch is studied. Quantum effects are calculated for the spectrum of radiated photons. Using this spectrum, the dependence of the relative energy loss δ on a quantum parameter K is discussed. It is shown that the behaviour of δ changes considerably with the increase of that parameter. In the classical regime (K ? 1) the energy loss is proportional to the incoming particle energy, while in the extreme quantum regime (K ? 1) the energy loss becomes a constant. The coherent e⁺e^? pair production for γe colliders as cross-channel to CBS is considered.