Identical motion in relativistic quantum and classical mechanics

The Klein-Gordon equation for the stationary state of a charged particle in a spherically symmetric scalar field is partitioned into a continuity equation and an equation similar to the Hamilton-Jacobi equation. There exists a class of potentials for which the Hamilton-Jacobi equation is exactly obtained and examples of these potentials are given. The partitionAnsatz is then applied to the Dirac equation, where an exact partition into a continuity equation and a Hamilton-Jacobi equation is obtained.     

Application of the complex point source method to the Schrödinger equation

The paraxial wave equation is a reduced form of the Helmholtz equation. Its solutions can be directly obtained from the solutions of the Helmholtz equation by using the method of complex point source. We applied the same logic to quantum mechanics, because the Schrödinger equation is parabolic in nature as the paraxial wave equation. We defined a differential equation, which is analogous to the Helmholtz equation for quantum mechanics and derived the solutions of the Schrödinger equation by taking into account the solutions of this equation with the method of complex point source. The method is applied to the problem of diffraction of matter waves by a shutter.     

Field theory onR×S 3 topology. I: The Klein-Gordon and Schrödinger equations

A Klein-Gordon-type equation onR×S ³ topology is derived, and its nonrelativistic Schrödinger equation is given. The equation is obtained with a Laplacian defined onS ³ topology instead of the ordinary Laplacian. A discussion of the solutions and the physical interpretation of the equation are subsequently given, and the most general solution to the equation is presented.     

An operator splitting method for the Degasperis–Procesi equation

An operator splitting method is proposed for the Degasperis–Procesi (DP) equation, by which the DP equation is decomposed into the Burgers equation and the Benjamin–Bona–Mahony (BBM) equation. Then, a second-order TVD scheme is applied for the Burgers equation, and a linearized implicit finite difference method is used for the BBM equation. Furthermore, the Strang splitting approach is used to construct the solution in one time step. The numerical solutions of the DP equation agree with exact solutions, e.g. the multipeakon solutions very well. The proposed method also captures the formation and propagation of shockpeakon solutions, and reveals wave breaking phenomena with good accuracy.     

Off-Shell Relativistic Quantum Mechanics and Formulation of Dirac's Equation Using Characteristic Matrices

It is argued that the Klein-Gordon equation isan equation for characteristic functions, i.e.,Fourier-transformed Wigner functions, not for wavefunctions. This statement is derived starting from theoff-shell formulation of relativistic quantum mechanicsby expressing the condition that the mass of theparticle is exactly known. A particular class ofsolutions of the Klein–Gordon equation is formedby the integrable superpositions of pure momentum states. Adirect sum of four copies of the associatedGelfand–Naimark–Segal representation isconsidered. Then one can derive from the Klein-Gordonequation an equation for spinor wave functions. Solutions of the latterequation are in one-to-one correspondence to thesolutions of the Fourier-transformed Dirac equation.Finally, the equation is reformulated as an equation for characteristic matrices.     

Master Equation and Fokker–Planck Equation: Comparison of Entropy and of Rate Constants

A usual approximation of the master equation is provided by the Fokker–Planck equation. For chemical systems with one species, we prove generally that the prediction of the rate constant of the metastable state given by the Master equation and the Fokker–Planck approximation differ exponentially with respect to the size of the system. We show that this is related to the fact that the entropy of the metastable state is not described correctly by the Fokker–Planck equation. We prove that the rate given by the Fokker–Planck equation overestimates that rate given by the Master equation.     

The modified propagation equation for TM polarized subwavelength spatial solitons in a nonlinear planar waveguide

The wave equation of TM polarized subwavelength beam propagations in a nonlinear planar waveguide is derived beyond the paraxial approximation. This modified equation contains more higher-order linear and nonlinear terms than the nonlinear Schrödinger equation. The propagation of fundamental subwavelength spatial solitons is numerically studied. It is shown that the effect of the higher nonlinear terms is significant. That is, for the propagation of narrower beam the modified nonlinear Schrödinger equation is more suitable than the nonlinear Schrödinger equation.     

THE IMPROVED MURNAGHAN EQUATION

The improved Murnaghan equation is derived by integrating B(P)=-v(∂p/∂v)_T=B₀⁽¹⁾(P)·P and expanding B⁽¹⁾(P) and simllar function in series of (1-K) , where K is the compression ratio (K≡(v/v₀)^1/3. The new equation obtained is compared with Murnagnan equation, Keane equation and Birch equation.It is found that, the improved Murnaghan equation has better convergency in comparison with the Birch equation. The characteristics of the new equation and the reasons of its better convergency are discussed.     

Vector Representation of Interacting Dirac Equation

Using the Clifford algebra, a vectorial equation for the Dirac spinorial equation is constructed and the relation with the Klein—Gordon equation becomes transparent. The equation interacting with the electromagnetic field leads to a nontrivial generalization for the interacting Klein—Gordon equation. The Lagrangian density for this interaction is given.     

Interaction of nonlinearity and polarization mode dispersion

The derivation of the coupled nonlinear Schrödinger equation and the Manakov-PMD equation is reviewed. It is shown that the usual scalar nonlinear Schrödinger equation can be derived from the Manakov-PMD equation when polarization mode dispersion is negligible and the signal is initially in a single polarization state as a function of time. Applications of the Manakov-PMD equation to studies of the interaction of the Kerr nonlinearity with polarization mode dispersion are then discussed.     

A discussion of the relativistic equal-time equation

Ruan Tu-nan et al. [1] have proposed an equal-time equation for composite particles which is derived from Bethe-Salpeter (B-S) equation. Its advantage is that the kernel of this equation is a completely definite single rearrangement of the B-S irreducible kernel without any artificial assumptions. In this paper we shall give a further discussion of the properties of this equation. We discuss the behaviour of this equation as the mass of one of the two particles approaches the limitM ₂→∞ in the ladder approximation of single photon exchange. We show that up to orderO(α⁴) this equation is consistent with the Dirac equation. If the crossed two photon exchange diagrams are taken into account the difference between them is of orderO(α⁶).     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

Fast sweeping method for the factored eikonal equation

We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part. The key idea is to enforce the causality of the original eikonal equation during the Gauss–Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation, i.e., the number of iterations for the Gauss–Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation, especially for point sources.     

ANALYTICAL SOLUTION OF THE 2n-DIMENSIONAL FOKKER-PLANCK EQUATION WITH HARMONIC OSCILLATOR POTENTIAL

An analytical solution is obtained for the 2n-dimensiona Fokker-Planck equation (F-P equation for short) with the harmonic oscillator potential. A few steps are involved in the derivation. First,the Lagrangian subsidiary equation is solved; then with its integral constants as new variables of the F-P equation, the diffusion equation is obtained and solved; at last, expressed in the original phase space, the solution of the F-P equation .is finally obtained. The analysis for the solution is made. The solution is a Gaussian type function and a δ-function of time. If a particle moves in a well in ali directions, then as t→∞, the distribution function can reach a stationary nonzero distribution-Maxuwell-Boltzmann type distribution (M-B distribution for short).As an example, the 2-dimensional F-P equation is solved and discussed in detail.     

A New Approach to the Exact Solutions of the Effective Mass Schrödinger Equation

Effective mass Schrödinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrödinger equation starting from the constant mass Schrödinger equation.     

Exact solutions of the generalized Bretherton equation

The generalized Bretherton equation is studied. The Bäcklund transformations between traveling wave solutions of the generalized Bretherton equation and solutions of polynomial ordinary differential equation are constructed. The classification problem for meromorphic solutions of the latter equation is discussed. Several new families of exact solutions for the generalized Brethenton equation are given.     

Separation of variables in the two-point BBGKY equations

Under two particular closure conditions, the two-point BBGKY equation is shown to be separable into equations for one- point turbulent fluctuations, yielding, respectively, a linear equation anda nonlinear integro-differential equation of convolution type. Analogy with Schrödinger's equation is discussed.     

Classification of exactly solvable potential problems

A differential equation with a known solution is transformed by changing both its dependent and independent variables, and the resulting nonlinear differential equation is then compared with the Schrödinger equation. The method is demonstrated using the confluent hypergeometric differential equation and the solutions to hydrogen, SHO and l=0 Morse potential problems are obtained.     

Propagation-Dispersion Equation

A propagation-dispersion equation is derived for the first passage distribution function of a particle moving on a substrate with time delays. The equation is obtained as the hydrodynamic limit of the first visit equation, an exact microscopic finite difference equation describing the motion of a particle on a lattice whose sites operate as time-delayers. The propagation-dispersion equation should be contrasted with the advection-diffusion equation (or the classical Fokker–Planck equation) as it describes a dispersion process in time (instead of diffusion in space) with a drift expressed by a propagation speed with non-zero bounded values. The temporal dispersion coefficient is shown to exhibit a form analogous to Taylor's dispersivity. Physical systems where the propagation-dispersion equation applies are discussed.     

Representation of solutions of the Burgers equation using functional integrals

In the paper, a representation of a solution of the Burgers equation in ℝ ⁿ is obtained by using integrals with respect to the Wiener measure on the space of trajectories in ℝ ⁿ . The Burgers equation is considered in a rigged Hilbert space. It is proved that, in the infinite-dimensional case, there is an analog of the Cole-Hopf transformation relating the Burgers equation and an analog of the heat equation with respect to measures. The Feynman-Kac formula for the heat equation (with potential) with respect to measures in a rigged Hilbert space is obtained.