VPT2+K spectroscopic constants and matrix elements of the transformed vibrational Hamiltonian of a polyatomic molecule with resonances using Van Vleck perturbation theory

Vibrational levels of polyatomic molecules are analysed with Van Vleck perturbation theory to connect experimental energy levels to computed molecular potential energy surfaces. Vibrational matrix elements are calculated from a quartic potential function via second-order Van Vleck perturbation theory, a procedure that treats both weak and strong interactions among vibrational states by approximately block-diagonalising the vibrational Hamiltonian. A clear and complete derivation of anharmonic and resonance constants as well as general expressions for both on- and off-diagonal matrix elements of the transformed Hamiltonian is presented. The equations are written in partial fraction form and as a constant multiplied by a harmonic oscillator matrix element to facilitate removing the effect of strongly interacting resonant states both in analytical formulae and in computer code. The derived equations are validated numerically, and results for the isotopomers of formaldehyde (H₂CO, HDCO, D₂CO) are included. The implications of the equations on zero-point energy calculations and experimental fits are discussed. The VPT2+K method is defined by these results for use in fitting and calculating vibrational energy levels.     

An empirical approach to the theory of particle and nuclear phenomena: Review and some new ideas

Experimental data on masses and lifetimes of unstable particles falls into a pattern, a brief review of some interesting consequences is presented here. From the experience in semiclassical methods and recent advances in quantum chromodynamics, it is proposed that an appropriate generalization of the Gutzwiller trace formula for field theories may lead to a systematic semiclassical chromodynamics theory. The theory can be developed to get appropriate dynamics leading to an explanation of pattern discovered in the empirical data.     

The geometric phase and ray space isometries

We study the behaviour of the geometric phase under isometries of the ray space. This leads to a better understanding of a theorem first proved by Wigner: isometries of the ray space can always be realised as projections of unitary or anti-unitary transformations on the Hilbert space. We suggest that the construction involved in Wigner’s proof is best viewed as an use of the Pancharatnam connection to ‘lift’ a ray space isometry to the Hilbert space.     

Van der Waals density functional theory study for bulk solids with BCC,FCC, and diamond structures

Proper inclusion of van der Waals (vdW) interactions in theoretical simulations based on standard density functional theory (DFT) is crucial to describe the physics and chemistry of systems such as organic and layered materials. Many encouraging approaches have been proposed to combine vdW interactions with standard approximate DFT calculations. Despite many vdW studies, there is no consensus on the reliability of vdW methods. To help further development of vdW methods, we have assessed various vdW functionals through the calculation of structural properties at equilibrium, such as lattice constants, bulk moduli, and cohesive energies, for bulk solids, including alkali, alkali-earth, and transition metals, with BCC, FCC, and diamond structures as the ground state structure. These results provide important information for the vdW-related materials research, which is essential for designing and optimizing materials systems for desired physical and chemical properties.     

The Mathai-Quillen formalism and topological field theory

These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the Mathai-Quillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the Atiyah-Jeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections.     

Local structure theory: Calculation on hexagonal arrays,and interaction of rule and lattice

Local structure theory calculations⁷ are applied to the study of cellular automata on the two-dimensional hexagonal lattice. A particular hexagonal lattice rule denoted (3422) is considered in detail. This rule has many features in common with Conway'sLife. The local structure theory captures many of the statistical properties of this rule; this supports hypotheses raised by a study ofLife itself⁽⁶⁾. As inLife, the state of a cell under (3422) depends only on the state of the cell itself and the sum of states in its neighborhood at the previous time step. This property implies that evolution rules which operate in the same way can be studied on different lattices. The differences between the behavior of these rules on different lattices are dramatic. The mean field theory cannot reflect these differences. However, a generalization of the mean field theory, the local structure theory, does account for the rule-lattice interaction.     

The double slit experiment,the Aharonov-Bohm effect,and a new quantum potential theory: Towards a realistic interpretation of quantum mechanics

In this paper, a new proposal of the quantum potential theory is presented, along with a discussion of the two-slit interference experiment and the Aharonov-Bohm effect, which are explicitly calculated within the scheme adopted. The present formulation holds the notion of particle trajectory and follows the interpretation suggested by de Broglie and Bohm, the quantum potential being a manifestation of the quantum wave. This wave always satisfies the Schrödinger equation and is considered as areal and separate entity from the particle with which is associated. In our approach we try to circumvent some conceptual difficulties that prevent the de Broglie-Bohm's theory from fulfilling a complete objective program. New effects are predicted, specially for the Aharonov-Bohm experiment in the regime of high magnetic fields, and a setup for their detection is proposed.     

Lie-optics, geometrical phase and nonlinear dynamics of self-focusing and soliton evolution in a plasma

It is useful to state propagation laws for a self-focusing laser beam or a soliton in group-theoretical form to be called Lie-optical form for being able to predict self-focusing dynamics conveniently and amongst other things, the geometrical phase. It is shown that the propagation of the gaussian laser beam is governed by a rotation group in a non-absorbing medium and by the Lorentz group in an absorbing medium if the additional symmetry of paraxial propagation is imposed on the laser beam. This latter symmetry, however, needs care in its implementation because the electromagnetic wave of the laser sees a different refractive index profile than the laboratory observer in this approximation. It is explained how to estimate this non-Taylor paraxial power series approximation. The group theoretical laws so-stated are used to predict the geometrical or Berry phase of the laser beam by a technique developed by one of us elsewhere. The group-theoretical Lie-optic (or ABCD) laws are also useful in predicting the laser behavior in a more complex optical arrangement like in a laser cavity etc. The nonlinear dynamical consequences of these laws for long distance (or time) predictions are also dealt with. Ergodic dynamics of an ensemble of laser beams on the torus during absorptionless self-focusing is discussed in this context. From the point of view of new physics concepts, we introduce a stroboscopic invariant torus and a stroboscopic generating function in classical mechanics that is useful for long-distance predictions of absorptionless self-focusing.     

Trigonometrical sums connected with the chiral Potts model,Verlinde dimension formula,two-dimensional resistor network,and number theory

We have recently developed methods for obtaining exact two-point resistance of the complete graph minus N$N$ edges. We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice, in proving Scott’s conjecture on permanent of Cauchy matrix, and in the perturbative chiral Potts model. The generalized trigonometrical sums of the chiral Potts model are shown to satisfy recursion formulas that are transparent and direct, and differ from those of Gervois and Mehta. By making a change of variables in these recursion formulas, the dimension of the space of conformal blocks of SU(2)$SU\left(2\right)$ and SO(3)$SO\left(3\right)$ WZW models may be computed recursively. Our methods are then extended to compute the corner-to-corner resistance, and the Kirchhoff index of the first non-trivial two-dimensional resistor network, 2×N$2\times N$. Finally, we obtain new closed formulas for variant of trigonometrical sums, some of which appear in connection with number theory.     

黑体辐射公式的多种推导及其在近代物理构建中的意义(Ⅰ)

黑体辐射是近代物理史上一只会下金蛋的鹅,是近代物理的摇篮。黑体辐射研究的意义还在于这是唯一一个涉及c,k,h三个普适常数的物理情景。黑体辐射谱抗测量误差的特性带来了辐射标准和绝对温度参照,谱分布公式对模型的不敏感则使得黑体辐射成为独特的物理研究母题。黑体辐射谱分布公式,普朗克多角度推导过,德拜推导过,艾伦菲斯特推导过,劳厄推导过,洛伦兹和庞加莱深入讨论过,泡利推导过,玻色推导过,爱因斯坦在20多年的时间里多角度推导过且产出最为丰硕,近代还有从相对论角度的推导,每一个角度的推导都带来了物理学的新内容,这包括量子力学、固体量子论、受激辐射、量子统计、相对论统计,等等。认真回顾黑体辐射研究的历史细节,考察其中的思想概念演化,不啻于体验一次教科书式的学(做)物理之旅,比如也可以尝试给出能量局域分立化的简单新证明。     

Random quantum states: recent developments and applications

We review the methods and use of random quantum states with particular emphasis on recent theoretical developments and applications in various fields. The guiding principle of the review is the idea that random quantum states can be understood as classical probability distributions in the Hilbert space of the associated quantum system. We show how this central concept connects questions of physical interest that cover different fields such as quantum statistical physics, quantum chaos, mesoscopic systems of both non-interacting and interacting particles, including superconducting and spin–orbit phenomena, and stochastic Schrödinger equations describing open quantum systems.