One-dimensional integral equations for a system of three identical particles in the boundary condition model and the possibility of changing the off-shell behaviour of the two-particle T-matrix

It is shown that in the framework of the boundary condition model (BCM) for the two-particle interaction the Schrödinger equation for the system of three identical bosons can be reduced to the one-dimensional integral equation in an exact way. The method used for obtaining such an equation is based on a special consideration of the two-particle off-shell wave functions. The binding energy of the simple three-particle system is calculated. It is indicated that by means of the equation obtained it is possible to change the off-shell behaviour of the two-particle (-matrix and therefore to simulate three-particle effects.     

A non‐uniqueness problem of the Dirac theory in a curved spacetime

The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. The latter equations may be defined when the spacetime is four‐dimensional, noncompact, and admits a spinor structure. We find that, for each among the three versions of the equation, the vast majority of the possible coefficient changes do not lead to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. In particular, we prove that the Dirac energy spectrum is not unique. This non‐uniqueness of the energy spectrum comes from an effect of the choice of coefficients, and applies in any given coordinates.     

Quantum-mechanical substantiation of a similar change in the kinetic energy component of an electron along the magnetic field within each Landau level

A detailed solution to the Schr?dinger equation for an electron in the presence of a quantizing magnetic field is considered using the method of separation of variables. The discrete energy levels occurring in this case are regarded as the potential barriers of finite height, with an electron moving normal to them along the magnetic-field direction. A similar change in the difference between the energy values within the neighboring Landau levels allows us to show, basing on the general properties of one-dimensional electron motion in the magnetic field, that within each quantum energy level, the kinetic-energy component of a longitudinal electron motion continuously is varied between 0 and 2μ_BH, whereas the probability density of particle location is practically damped into the potential-barrier depth as it moves along the magnetic-field direction. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 3–7, October, 2006.     

Theoretical Study of the Energies of the Oscillating System with a Well- Distributed Mass of the Spring

The energy of a spring with a well-distributed mass m_s is theoretically studied in this paper. The solution of the wave equation is derived in detail, and then the kinetic energy and potential energy of the spring are studied with the wave equation, as well as the kinetic energy of the oscillating mass M. The kinetic energy and potential energy of the spring, and total energy are numerically simulated for different ratios m_s/M with considering the spring’s mass, which makes the property of energy of the oscillating system understood easily.     

Quantum dynamics in a time-dependent cylindrical trap

Solutions to the Schrödinger equation are examined for a particle inside a cylindrical trap of a circular time-dependent cross-section. Analytical expressions for energy and momentum expectation values are derived with respect to the exact solutions; and the adiabatic and sudden change of the boundary are discussed. The density profile as a function of time in a given observation point, resembles the diffraction-in-time pattern observed in a suddenly released particle but with an enhanced fringe visibility. Numerical computations are presented for both contracting and expanding boxes.     

Remarks on the electromagnetic waves in a medium with negative ɛ and µ

The question whether main electrodynamics laws accept the existence of materials with negative ? and µ is considered. An obstacle to this is the negativity of the wave energy density W in the theory of such media. It is shown that an attempt to change W sign only worsens the situation since it leads to the conflict of signs in the continuity equation.     

Viscous brane cosmology with a brane-bulk energy interchange term

We assume a flat brane located at y = 0, surrounded by an AdS space, and consider the 5D Einstein equations when the energy flux component of the energy-momentum tensor is related to the Hubble parameter through a constant Q. We calculate the metric tensor, as well as the Hubble parameter on the brane, when Q is small. As a special case, if the brane is tensionless, the influence from Q on the Hubble parameter is absent. We also consider the emission of gravitons from the brane, by means of the Boltzmann equation. Comparing the energy conservation equation derived herefrom with the energy conservation equation for a viscous fluid on the brane, we find that the entropy change for the fluid in the emission process has to be negative. This peculiar effect is related to the fluid on the brane being a non-closed thermodynamic system. The negative entropy property for non-closed systems is encountered in other areas in physics also, in particular, in connection with the Casimir effect at finite temperature.     

Spectrum of stationary electromagnetic radiation in a static gravitational field

From the equation of continuity, an equation is derived for the change in the spectrum of radiation stationary in a static gravitational field, as measured by local observers in their proper frame. It is shown that an isotropic solution always exists in which the energy density per unit frequency interval is ³ times an arbitrary function of .     

Transport phenomena in a capillary-porous medium I. The model of a capillary-porous body and a series of balance equations

The reality of a capillary-porous body is substituted by the model of a continuous viscoelastic, isotropic two-component medium (component 1 is the solid phase, component 2 is a liquid). It has been proved that if a polymolecular adsorbent layer is created, this model is physically adequate to the reality. In this paper, the balance equation of mass, the momentum, the moment of momentum, the kinetic energy, the total energy, and the internal energy are derived for this model. Since it is characteristic of an arbitrarily chosen volume of the investigated system to change its mass, the derivation of these equations is based on Me??erski's instead of on Newtonian mechanics as is usual. The binding energy, which characterizes the energy interaction in sorbent processes, is considered. Tensors are found that describe the swelling of the investigated body and the friction between the fluid and the solid phase. The results are generalized for systems confined within diathermic, soft, unscreening and semipermeable walls.     

Two critical field strengths in the Copenhagen vacuum

One-loop corrections to the equation expressing the instability of a constant colour magnetic field with respect to the Nielsen-Olesen mode are considered. These corrections radically change the equation, and introduce a critical field strength B_C upon which instability, and consequently the life or death of the Copenhagen vacuum, depends. The value of B_C is calculated, and is found to leave the Copenhagen vacuum intact. Our equation distinguishes between field configurations, which are quantum dynamically unstable and configurations having an effective energy with non-zero imaginary part, generally believed to signal instability. We interpret the difference. Finally we expose a mechanism which presumably makes the imaginary part of the effective energy density vanish, if the Nielsen-Olesen unstable mode is excited above a critical amplitude χ_C ≈ 0.24.     

Coordinate-dependent 3+1 formulation of the general relativity equations for a perfect fluid

This paper defines mass, momentum, and energy densities for a perfect fluid, and derives a coordinate-dependent 3+1 decomposition of the equation of motion in terms of a scalar potential c ² [(–g _g44) 1/2 –1] and a vector potentialA _i cg _4i /(–g ₄₄)^1/2. The momentum equation has the form of the Euler equation except there is an additional force proportional to the vector potential and the rate of change of kinetic energy per unit volume. The momentum and energy equations are integrated to obtain the equations previously derived for a particle. The momentum equation is solved for the total acceleration of a fluid element. The equations are exact and do not depend on the choice of coordinate system.     

Calculation of radiant energy in free electron lasers without inversion

The equations of motion of particles in free electron lasers without inversion (FELWI) were obtained using the Hamilton formalism. An one-dimensional decoupled phase equation of the type of mathematical pendulum equation was derived in the case of weak signal. For practical estimates, this equation was solved together with the equation of electron energy change within the frameworks of perturbation theory, and expressions for gain were obtained in FEL regime and for the output angle from the first undulator. Expressions for the gain coefficient, the phase and the energy change were found as functions of the parameters of beam and of the device.     

The accommodation of the translational and rotational energy of a gas in a Knudsen flow past a thin wire

The paper presents the results obtained in determining the accommodation coefficients for the translational and rotational energy of gas molecules in a Knudsen flow past a thin wire. The method used was based on numerically solving the complete heat balance equation for a wire probe. The accommodation coefficients were determined for H₂, N₂, CH₄, and CO₂ on a gilded tungsten surface. For hydrogen with a quenched rotational energy, a negative accommodation coefficient of rotational energy was obtained due to the conversion of the rotational energy of incident molecules into the translational energy of reflected molecules.     

Solutions of the Fokker-Planck equation describing the thermalization of neutrons in a heavy gas moderator

The thermalization of neutrons is described by a transport equation with a second order differential operator with respect to the energy. First this equation is transformed to an one-variable Fokker-Planck equation. Next an eigenfunction expansion and a polynomial expansion are used to solve the time-dependent Fokker-Planck equation. The eigenvalues are obtained either by solving a Schrödinger equation or by calculating 2×2 matrix continued fractions. Explicit results for the approach to equilibrium of a pulse of neutrons as well as the stationary distribution for 1/v absorption are presented. It is shown that the theory of the photoelectromotive force in semiconductors also leads to the same problem.     

On the prediction of impact noise,III: Energy accountancy in industrial machines

It has been explained in the two previous parts of this series of papers that the peak noise level of an impact machine can be predicted in terms of the initial kinetic energy in an equivalent bag of air of the same dimensions and impact time, but that the total L_eq at any frequency must depend upon the vibrational energy stored in the machine following impact and in its ability to radiate such energy as sound. This paper presents a method of examining the energy balance in the machine and obtains simple and helpful expressions for the radiated sound energy in terms of the internal energy level, a “modified” radiation efficiency, a structural damping factor, and a bulkness factor (equation (4a)). An alternative form, based upon the measured rates of change of force in the operating tool, impulse and structural response factors, modified radiation efficiency, damping factor and bulkiness factor is also presented (equation (4b)). The diagnostic advantages of thinking in terms of the changes in each of these terms as a method of studying noise control, even for apparently continuous running processes, are illustrated by a series of case studies which include punch press analysis, bottle impacts, impacts of solid workpieces falling into bins, gearwheel noise phenomena, backlash and diesel engine noise.     

南海海域跨海沟环境的声场会聚特性

利用南海北部海域的声传播实验数据分析了跨海沟条件下的声传播特性。结果表明:由于负梯度声速剖面和海底地形的共同作用,导致声能量在开始随着海沟深度变化向更深层位置上弯曲传播,传播损失在海沟中心位置附近达到最大。在海深逐渐变浅的距离上,由于海底反射使得声能量逐渐会聚,声传播损失比单纯陆坡变到深海环境下要减小20 dB以上。当会聚效果小于扩散和海底反射等引起的损失时,声传播损失达到最小,之后随着距离增大而增大;在海沟最后一段,当海深在8 km范围内从850 m突然变浅到311 m,声场能量逐渐会聚效果再次凸显,使得声传播损失减小10 dB。利用射线理论和抛物方程近似数值分析,解释了海沟环境下的由于地形变化引起的声场会聚传播特性。      

Formation of the charge and energy distributions of heavy ions in a material in the diffusion model

We investigate the formation of the charge and energy distributions of ions that slow down and randomly change their charges in collisions with particles of the medium. We study the influence that the spread of ions in charge has on the shape of the Bragg curve. The suggested diffusion approximation for the kinetic equation of heavy ions allows the parameters of the ion charge and energy distributions to be easily determined in the entire ion path. The parameters of the ion charge distribution are shown to be related to the ionization-recombination cross sections. The ion distributions calculated in the suggested analytical model are compared with the results of numerical calculations. We have obtained good agreement between the analytical, numerical, and experimental results.     

Thermodynamics in the statistical theory of a crystal

The thermodynamics of a crystal is developed on the basis of the Vlasov equation. From this equation a system of equations is derived for the Fourier coefficients of the density. The internal energy and stress tensor are found. It is shown that the results obtained satisfy the second law of thermodynamics and the entropy is calculated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 85–88, December, 1978.     

Coordinate-dependent 3 + 1 formulation of the general relativity equation of motion

This paper gives a coordinate-dependent 3 + 1 decomposition of the geodesic law of motion. The equation is similar in form to the equation of motion of electrodynamics except there is an extra term proportional to the rate of change of kinetic energy and the vector potential. The equation is solved for the acceleration explicitly.     

Resonances in the one-dimensional Dirac equation in the presence of a point interaction and a constant electric field

We show that the energy spectrum of the one-dimensional Dirac equation in the presence of a spatial confining point interaction exhibits a resonant behavior when one includes a weak electric field. After solving the Dirac equation in terms of parabolic cylinder functions and showing explicitly how the resonant behavior depends on the sign and strength of the electric field, we derive an approximate expression for the value of the resonance energy in terms of the electric field and delta interaction strength.