Hydrogen atom in quantum mechanics and quantization on curved surfaces

New quantization rules for classical systems are obtained using the Titchmarsh expansion. These rules generalize the conventional ones and are reduced to them when a transition to Cartesian coordinates exists. An equation generalizing the Schrödinger equation to arbitrary natural systems is found. The principle of minimal constraint (strong equivalence principle) makes it possible to extend this equation to any curved spaces.     

On the uniqueness of harmonic coordinates

Harmonic coordinate conditions in stationary asymptotically flat spacetimes with matter sources have more than one solution. The solutions depend on the degree of smoothness of the metric and its first derivatives, which we wish to impose across the material boundary, and on the conditions at infinity and at a suitable point inside the matter. This is illustrated in detail by simple fully solvable examples of static spherically symmetric spacetimes in global harmonic coordinates. Examples of stationary electrovacuum spacetimes described simply in harmonic coordinates are also given. They can represent the exterior fields of material discs.The use of an appropriate background metric considerably simplifies the calculations.Dedicated to Prof. Jirí Horáek on the occasion of his 60th birthday Undoubtedly, Jirí has been using harmonic coordinates — in flat space these are the Cartesian coordinates — in his quantum calculations happily for years. We hope that his sense for harmony in music and life will also appreciate the harmonic gauge in curved spacetimes ....     

Application of the method of kinemetric invariants to the Cauchy problem in general relativity

A class of metrics in which the spatial part describes flat space is considered. The algebraic condition of reduction of the three-dimensional part of an arbitrary metric to Cartesian form is found. The Cauchy problem for these metrics is considered in terms of kinemetrically invariant quantities. The results are used to solve the Cauchy problem for a spherically symmetric gravitational field. A solution is also obtained for the tachyon twin of the Schwarzschild field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 48–52, April, 1976.     

Path Integration and Separation of Variables in Spaces of Constant Curvature in Two and Three Dimensions

In this paper path integration in two- and three-dimensional spaces of constant curvature is discussed: i.e. the flat spaces R² and R³, the two- and three-dimensional sphere and the two- and three-dimensional pseudosphere. We are going to discuss all coordinates systems where the Laplace operator admits separation of variables. In all of them the path integral formulation will be stated, however in most of them an explicit solution in terms of the spectral expansion can be given only on a formal level. What can be stated in all cases, are the propagator and the corresponding Green function, respectively, depending on the invariant distance which is a coordinate independent quantity. This property gives rise to numerous identities connecting the corresponding path integral representations and propagators in various coordinate systems with each other.     

Smorodinsky-Winternitz potentials: Coherent state approach

In this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional harmonic oscillator problem and the Kepler-Coulomb problem. In the first case we find the nonspreading wave packets by transforming the system into four oscillators in Cartesian, and also polar, coordinates. In the second case, the coherent states are constructed in Cartesian coordinates by transforming the system into three nonisotropic harmonic oscillators. All of these states evolve in physical-time. In the third case, the system is transformed into four oscillators and the parametric-time coherent states are constructed in two coordinate frames. In the fourth case, the system is transformed into two oscillators with the reflection symmetry and the parametrictime coherent states are constructed in two coordinate frames.     

Generalized Robertson-Walker metrics and some of their properties. II

The Robertson-Walker (RW) metrics, of dimensionality four and signature –2, are generalized to metrics of dimensionality (n+1) and of arbitrary signature,n (> 1) being an arbitrary integer. In canonical coordinates (t, x ¹,x ², ...,x ⁿ ) these generalized Robertson-Walker (GRW) metrics are functions of the coordinatet. The following statements are proved to be equivalent: The GRW metrics are (a) expressible int-independent form, (b) of constant curvature, (c) Einstein spaces. Furthermore, there are six, and only six, classes of GRW metrics satisfying these three statements. The coordinate transformations which transform these metrics to theirt-independent form are given explicitly. Two of these classes of GRW metrics reduce, in theirt-independent form, to the same flat (generalized Minkowski) metrics, three reduce to the samet-independent metrics which are generalizations of the de Sitter space-time metric, and the last class tot-independent metrics which are generalizations of the anti-de Sitter space-time metric.     

A new approach to obtaining sum rules

A new approach to obtaining the sum rules satisfied by the phenomenological coefficients appearing in the metric tensor of the generalized coordinate space of a many-particle system is considered and applied to the vibration-rotation motion of a nonlinear molecule. The approach is based on the requirement of physical covariance of the Cartesian and generalized coordinate configuration spaces of the system. A partial criterion of this physical covariance is the condition that the curvature tensor of configuration space remains covariant, and this condition gives several sum rules for the vibration-rotation parameters. These sum rules are particular cases of those known previously.     

Sagnac Effect of Gödel's Universe

We present exact expressions for the Sagnac effect of Gödel's Universe. For this purpose we first derive a formula for the Sagnac time delay along a circular path in the presence of an arbitrary stationary metric in cylindrical coordinates. We then apply this result to Gödel's metric for two different experimental situations: First, the light source and the detector are at rest relative to the matter generating the gravitational field. In this case we find an expression that is formally equivalent to the familiar nonrelativistic Sagnac time delay. Second, the light source and the detector are rotating relative to the matter. Here we show that for a special rotation rate of the detector the Sagnac time delay vanishes. Finally we propose a formulation of the Sagnac time delay in terms of invariant physical quantities. We show that this result is very close to the analogous formula of the Sagnac time delay of a rotating coordinate system in Minkowski spacetime.     

曲面上的动量和动能算符

对约束在曲面上粒子运动的描述可以在内部坐标即曲面局部坐标下进行，也可以在外部坐标即在笛卡尔坐标下进行.在量子力学中，动量和动能算符的表示在这两种描述中各有不同，前者的动量算符仅包含内禀几何量，后者的动量算符包含了曲面的平均曲率.考虑到算符次序问题，动能算符对动量算符的依赖关系也不同，前者的依赖关系仅发现存在一种，后者的依赖关系已经发现有两种.     

曲面上的动量和动能算符

对约束在曲面上粒子运动的描述可以在内部坐标即曲面局部坐标下进行,也可以在外部坐标即在笛卡尔坐标下进行.在量子力学中,动量和动能算符的表示在这两种描述中各有不同,前者的动量算符仅包含内禀几何量,后者的动量算符包含了曲面的平均曲率.考虑到算符次序问题,动能算符对动量算符的依赖关系也不同,前者的依赖关系仅发现存在一种,后者的依赖关系已经发现有两种. 关键词： 量子力学 微分几何     

On Gauge Invariant Theories of Gravitation

Theories of gravitation are called gauge invariant if the invariance of the gravitational field lagrangian with respect to gauge transformations of the gravitational field variables is independend of the invariance of this lagrangian with respect to the Einstein group of general coordinate transformations. They are bimetric theories because the coordinate covariance is ensured by constructing scalar densities relative to a globally flat background metric. Such a theory is represented by the PAUL-FIERZ equations for massless spin 2 particles. But this theory is inconsistent if nongravitational matter is enclosed as a source. All attempts to overcome this inconsistancy preserving gauge invariance lead to Einstein's GRT. We review this problem and compare the situation with a theory proposed by LOGUNOV showing that he overcomes the inconsistency of linear Einstein's equations by replacing the field variables by a gauge invariant combination of new ones, which turns out to be the first order form of v. FREUD'S superpotential.     

Power flow of the guided mode of a waveguide

The integral ∫_ΩΦΨdΩ taken over an arbitrary plane region Ω where the scalar functions of the point Φ and Ψ are the solutions of the Helmholtz two-dimensional equation is presented as a contour, i.e., in the invariant view and in three main orthogonal coordinate systems on a plane, namely, in the Cartesian, polar, and elliptic coordinate systems. An invariant expression in the view of the contour integral for power flow of the guided mode through an arbitrary region of the cross section of a waveguide with constant permittivity has been obtained.     

Separation of variables in Einstein spaces. II. No ignorable coordinates

We prove that the only Einstein spaces which admit a coordinate system with no ignorable coordinates which separates the Hamilton-Jacobi equation are certain symmetric spaces of Petrov typeD due to Kasner and the constant-curvature de Sitter spaces. We also show that a space admitting a coordinate system with no ignorable coordinates which separates the Helmholtz (Schrödinger) equation must be of Petrov type     

Metric and Ricci tensors for a certain class of space-times ofD type

The class of space-times has been determined at the connection level, assuming the existence of some symmetrical relations between the Ricci rotation coefficients. It has been assumed, for instance, that at least two shear-free congruences of null geodesics exist. We have shown that onlyD type or conformally flat space-times can belong to this class. The theorem has been proved that a system of coordinates exists in which the metric tensor can depend on two coordinates, only. The metric tensor has been determined with an accuracy to two functions, each of which is a function of only one coordinate. Linear, second-order differential expressions have been found for these two functions. They determine the Ricci tensor. Several solutions of the Einstein-Maxwell equations with a cosmological constant are given.On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

A unifying coordinate family for the Kerr–Newman metric

A new unified metric form is presented for the Kerr–Newman geometry. The new form is a generalization of the Boyer–Lindquist metric involving an arbitrary gauge function of the spheroidal radial variable. Each choice of the gauge function corresponds to a coordinate system including four of the most important coordinate systems for Kerr–Newman (Boyer–Lindquist, Kerr, Kerr–Schild and Doran coordinates). The representation is given in terms of a single Minkowski frame together with the gauge function. This Minkowski frame arises by boosting a static orthonormal frame which is adapted to spheroidal coordinates. Properties of the boost reflect the rotating nature of the Kerr–Newman solution including an identification of the angular velocities of the disk and the horizon matching previously known values obtained in other ways.     

Eigenmodes for electromagnetic waves propagating in a toroidalcavity

A solution has been attempted by means of the Helmholtz equation for an electromagnetic wave propagating in an empty torus in a system of toroidal coordinates. The electromagnetic fields are expressed in terms of the Hertz vector to obtain a scalar Helmholtz equation. The latter has been solved by making use of an inverse aspect ratio expansion of the solution. Unlike most previous workers, the authors have obtained their solutions in terms of hypergeometric functions whose static limit is the toroidal harmonics. The cylindrical solutions in terms of Bessel functions can also be recovered by taking the appropriate large aspect ratio limit. The eigenmodes, with arbitrary toroidal and poloidal mode numbers, have been obtained by applying the boundary conditions on the metallic walls of infinite conductivity, and they cannot be distinguished as TE or TM modes. Eigenfrequencies for various toroidal and poloidal mode numbers are plotted against the inverse aspect ratio. First-order approximations to the fields in the toroidal cavity have also been derived     

Improving the resolution of three-dimensional acoustic imaging with planar phased arrays

This paper examines and compares two methods of improving the quality of three-dimensional beamforming with phased microphone arrays. The intended application is the detection of aerodynamic noise sources on wind turbines. Both methods employ Fourier based deconvolution. The first method involves a transformation of coordinates that tends to make the response to a point source, the point spread function, more shift invariant. The result is a significant improvement in sound source imaging in the transformed coordinate system. However, the inverse transformation to Cartesian coordinates introduces range dependent resolution limitations because of the irregular distribution of the focal points. The second method combines the transformation of coordinates with an alternative scanning technique. This method can be used in near field three-dimensional acoustic imaging to produce maps free of sidelobes and with constant resolution. The robustness of the proposed methods is validated both with computer simulations and experimentally.     

Special Stäckel electrovac spacetimes

Classification of all electrovac specetimes permitting the separation of variables in the Hamilton-Jacobi equation for a charged test particle is carried out. This separation requires the existence of a complete set consisting of Killing’s vectors and tensors of a special kind. Every complete set defines its own type of metric and electromagnetic potential in the separable coordinate system. There exist seven types of separation of variables for electromagnetic spaces. For every type an additional classification is carried out by transformation of coordinates without any disturbance of the separation conditions, the gradient transformation of electromagnetic potential and the conformal-constant transformation of metric. The key step in solving the problem is the extraction of an autonomous subsystem which determines the metric from only the Einstein-Maxwell equations for every type of separation of variables. Representatives of all classes of metrics and electromagnetic potential are given for every type of separation of variables with the exception of the spaces found in the well-known work by Carter. The problem is solved in terms of metric formalism. The classes of electrovac spacetimes obtained are found to be related to Petrov’s classification.