Uncertainty relation for photons

The uncertainty relation for the photons in three dimensions that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the energy distribution in space leads to an inequality that is a natural counterpart of the standard Heisenberg relation. The equation satisfied by the photon wave function in momentum space which saturates the uncertainty relations has the form of the Schr?dinger equation in coordinate space in the presence of electric and magnetic charges.     

Dynamics of one electron in a nonlinear disordered chain

In this paper we report new numerical results on the disordered Schr?dinger equation with nonlinear hopping. By using a classical harmonic Hamiltonian and the Su-Schrieffer-Heeger approximation we write an effective Schr?dinger equation. This model with off-diagonal nonlinearity allows us to study the interaction of one electron and acoustical phonons. We solve the effective Schr?dinger equation with nonlinear hopping for an initially localized wavepacket by using a predictor-corrector Adams-Bashforth-Moulton method. Our results indicate that the nonlinear off-diagonal term can promote a long-time subdiffusive regime similar to that observed in models with diagonal nonlinearity.     

环形非球谐振子标量势与矢量势的Klein-Gordon和Dirac方程的束缚态

通过求解环形非球谐振子势的 Klein-Gordon 和 Dirac方程,给出了束缚态波函数与能量方程的精确解,并指出非相对论Schr?dinger方程与具有相等标量势和矢量势的Klein-Gordon 方程以及Dirac方程之间具有数学结构上的等价性。     

Quantum-invariant theory and the evolution of a Dirac field in Friedmann–Robertson–Walker flat space-times

On the basis of the generalized invariant formulation, the invariant-related unitary transformation method is used to study the evolution of a quantum Dirac field in Friedmann–Robertson–Walker spatially flat space-times. We first solve the functional Schr?dinger equation for a free Dirac field and obtain the exact solutions. We then investigate the way of extending the method to treat the case in which there is an interaction between the Dirac field and a scalar field. Received: 17 July 1999 / Published online: 6 March 2000     

Covariant Quantization of Spinor Fields in a Given Gravitational Field

In coupling gravity with the quantum field theory, unitary transformations, depending on space-time-points, were considered and derivatives were introduced, which imply a nonintegrable parallel transport of the state vectors of Hilbert space [1]. The Dirac equation, built with these generalized derivatives, is quantized in a prescribed classical gravitational field. The quantization can be performed in complete analogy to the usual procedure in Minkowski space, but the quantum state vector becomes path dependent. In carrying out the quantization, two two-component classical spinor fields necessarily occur, which obey Weyl's equation. The considered quantized Dirac equations are also picture-covariant, that is they have the same from in each physical picture, especially in the Heisenberg picture and the Schrödinger picture.     

Quantum—classical correspondence of the time-dependent linear potential

Using the trial-function method, the general solution of the Schrödinger equation for the time-dependent linear potential is obtained. Based on the Heisenberg correspondence principle, the solution of the classical equation of motion is derived from the quantum matrix elements.     

On the Incompressible Fluid Limit and the Vortex Motion Law of the Nonlinear Schrödinger Equation

The nonlinear Schr?dinger equation (NLS) has been a fundamental model for understanding vortex motion in superfluids. The vortex motion law has been formally derived on various physical grounds and has been around for almost half a century. We study the nonlinear Schr?dinger equation in the incompressible fluid limit on a bounded domain with Dirichlet or Neumann boundary condition. The initial condition contains any finite number of degree ± 1 vortices. We prove that the NLS linear momentum weakly converges to a solution of the incompressible Euler equation away from the vortices. If the initial NLS energy is almost minimizing, we show that the vortex motion obeys the classical Kirchhoff law for fluid point vortices. Similar results hold for the entire plane and periodic cases, and a related complex Ginzburg–Landau equation. We treat as well the semi-classical (WKB) limit of NLS in the presence of vortices. In this limit, sound waves propagate through steady vortices. Received: 1 December 1997 / Accepted: 27 June 1998     

Time dependence in quantum mechanics

It is shown that the time-dependent equations (Schr?dinger and Dirac) for a quantum system can be derived from the time-independent equation for the larger object of the system interacting with its environment, in the limit that the dynamical variables of the environment can be treated semiclassically. The time which describes the quantum evolution is then provided parametrically by the classical evolution of the environment variables. The method used is a generalization of that known for a long time in the field of ion-atom collisions, where it appears as a transition from the full quantum mechanical perturbed stationary states to the impact parameter method in which the projectile ion beam is treated classically. Received 25 October 1999     

与不变量有关的幺正变换方法和含时均匀电场中的量子Dirac场的演化

在推广的不变量理论的基础上,运用与不变量有关的幺正变换方法研究了含时均匀电场下的量子Dirac场的演化,求解了Dirac场的泛函Schrdinger方程,得到了方程的精确解以及对应的总相位,总相位包括动力学相和几何相(Aharonov-Anandan phase). 关键词：     

Spectra of baryons containing two heavy quarks

Doubly heavy baryons, i.e., the baryons containing two heavy quarks are treated in the adiabatic approximation, considering the motion of the light quark as a relativistic motion. The binding energy and mass spectra of doubly heavy baryons are calculated solving the two-center Dirac equation the one-centre Schr?dinger equation for Coulomb plus linear potential. The binding energy of the light quark as a function of the distance between heavy quarks is found. Received: 16 January 2002 / Accepted: 7 February 2002     

一维定态薛定谔方程的宏观模拟解法

将实验模拟法引入量子力学.设计了一个弦振动系统,这个系统的定态方程与定态薛定谔方程数学形式一样,为定态薛定谔方程的模拟解法提供了理论和实验途径.宏观模拟的结果为理解薛定谔方程提供了宏观类比. 关键词： 薛定谔方程 宏观模拟解法     

Stationary dark localized modes: discrete nonlinear Schrödinger equations

Various kinds of stationary dark localized modes in discrete nonlinear Schr?dinger equations are considered. A criterion for the existence of such excitations is introduced and an estimation of a localization region is provided. The results are illustrated in examples of the deformable discrete nonlinear Schr?dinger equation, of the model of Frenkel excitons in a chain of two-level atoms, and of the model of a one-dimensional Heisenberg ferromagnetic in the stationary phase approximation. The three models display essentially different properties. It is shown that at an arbitrary amplitude of the background it is impossible to reach strong localization of dark modes. In the meantime, in the model of Frenkel excitons, exact dark compacton solutions are found.     

Perturbative aspects of q-deformed dynamics

Within the framework of the q-deformed Heisenberg algebra a dynamical equation of q-deformed quantum mechanics is discussed. The perturbative aspects of the q-deformed Schr?dinger equation are analyzed. General representations of the additional momentum-dependent interaction originating from the q-deformed effects are presented in two approaches. As examples, such additional interactions related to the harmonic-oscillator potential and the Morse potential are demonstrated. Received: 26 February 2001 / Published online: 11 May 2001     

Quantum Dynamics and Kinematics from a Statistical Model Selected by the Principle of Locality

Quantum mechanics predicts correlation between spacelike separated events which is widely argued to violate the principle of local causality. By contrast, here we shall show that the Schrödinger equation with Born’s statistical interpretation of wave function and uncertainty relation can be derived from a statistical model of microscopic stochastic deviation from classical mechanics which is selected uniquely, up to a free parameter, by the principle of Local Causality. Quantization is thus argued to be physical and Planck constant acquires an interpretation as the average stochastic deviation from classical mechanics in a microscopic time scale. Unlike canonical quantization, the resulting quantum system always has a definite configuration all the time as in classical mechanics, fluctuating randomly along a continuous trajectory. The average of the relevant physical quantities over the distribution of the configuration are shown to be equal numerically to the quantum mechanical average of the corresponding Hermitian operators over a quantum state.     

Relativistic New Yukawa-Like Potential and Tensor Coupling

We approximately solve the Dirac equation for a new suggested generalized inversely quadratic Yukawa potential including a Coulomb-like tensor interaction with arbitrary spin-orbit coupling quantum number ${\kappa}$ . In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions, in closed form, by using the parametric Nikiforov–Uvarov method. The numerical results show that the Coulomb-like tensor interaction, ?T/r, removes degeneracies between spin and p-spin state doublets. The Dirac solutions in the presence of exact spin symmetry are reduced to Schr?dinger solutions for Yukawa and inversely quadratic Yukawa potentials.     

Time-space fractional Schrödinger like equation with a nonlocal term

In this paper a time-space fractional Schr?dinger equation containing a nonlocal term has been studied. The time dependent solutions have been obtained in terms of the H-function. New general results include the results of integer Schr?dinger equation with a nonlocal term and the well-known quantum formulae for a free particle kernel.     

The time dependent Dirac-Frenkel-McLachlan variation of parameters: an NMR application

The time dependent variation of parameters solution to the time dependent Schr?dinger equation pioneered by Dirac, Frenkel, and McLachlan is described in terms useful for immediate application to complex time dependent problems in magnetic resonance. A benchmark comparison of the theory to one dimensional images and spin echo envelope signals in simple spatially varying magnetic fields with molecular diffusion is provided.     

Finite Temperature Schr?dinger Equation

We know Schr?dinger equation describes the dynamics of quantum systems, which don’t include temperature. In this paper, we propose finite temperature Schr?dinger equation, which can describe the quantum systems in an arbitrary temperature. When the temperature T=0, it become Shr?dinger equation.     

Superalgebras for three interacting particles in an external magnetic field

In this paper we discuss interacting particles in an external magnetic field. By comparing the Schr?dinger equation of three interacting particles with the associated Laguerre differential equation, we obtain the energy spectrum which corresponds to indices n_i and m_i. Finally by using the so called factorization method we obtain the raising and lowering operators. These operators are supersymmetric structures related to the Hamiltonian partner. Also these operators lead to the realization of Heisenberg Lie superalgebras with two, four and six supercharges.