Modified Perturbation Theory of an Anharmonic Oscillator

A modification of the perturbation theory of a symmetrical anharmonic oscillator is suggested. A more complex zero-order approximation of perturbation theory that considers to a certain degree anharmonicities is chosen rather than a harmonic oscillator model. This approximation is an analog of the self-consistent field model well known in the theory of many-particle systems. A comparison of modified and conventional perturbation theories demonstrates that the modified perturbation theory has much wider applicability range. It can be used for larger values of the parameters at which the conventional perturbation theory becomes inapplicable, namely, for strong anharmonicity and upper energy levels.     

On perturbation theory for an asymmetric anharmonic oscillator

A modified perturbation theory is formulated for an asymmetric anharmonic oscillator based on a choice of the main approximation constructed by analogy with the self-consistent field model. This perturbation theory has a much wider range of application in comparison with the standard approach and considers a finite number of energy levels in the potential well already in the main approximation. This approach is used for the construction of a two-atomic model of a quantum crystal. A quantum analog of the Lindeman criterion is obtained. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 30–40, January, 2009.     

Two-dimensional model of a kicked oscillator: Motion with intermittency

The phenomenon of intermittency, which arises near a point of degeneracy of an unperturbed Hamiltonian under the influence of a discontinuous perturbation function, is studied in the example of a two-dimensional (2-D) model of a kicked oscillator. This example describes the dynamics of a particle in a cylindrically symmetric potential well subjected to radial kicks which occur periodically in time. The problem is reduced to a Hamiltonian system with N=3/2 degrees of freedom, whose unperturbed Hamiltonian has a degeneracy point. The intermittency is studied numerically and analytically.     

二维极化子在磁场中的基态能量

谐振子算符的代数运算方法被用于研究磁场中同时与表面光学声子及表面声学声子相互作用的二维电子。得到二维极化子在强磁场中直至四级微扰的基态能量以及它在任意强经磁场中的二级微扰基态能量表达式。结果发现,对磁场中二维极化子基态能量的影响中,表面声学声子有着与表面光学声子同样的甚至更为突出的贡献,是不容忽视的。 关键词：     

Theoretical investigations of the vapour-liquid equilibrium and dielectric properties of dipolar Yukawa fluids in an external field

In a low field approximation, using the dipolar Yukawa fluid model (in mean spherical approximation as a reference system) a consistent field-dependent free energy expression is proposed for the calculation of the vapour-liquid equilibrium of polar fluids in an applied electric field. A perturbation theory high field approximation expression of the free energy is also proposed to study the field-dependent properties of fluids. In the high field approximation, equations for the field-dependent polarization and for the nonlinear dielectric constant (or Piekara constant) are also predicted. It has been discussed that our approximations are appropriate to describe the vapour-liquid-like phase equilibria and the magnetization curves of magnetic fluids.     

运用线性量子变换理论和定态微扰论求解线型多原子分子振动的一般方法

采用键长伸缩和垂直键轴位移为内坐标,多维耦合谐振子、势函数中超过二次幂的项分别为零级近似和微扰的线型多原子分子振动模型,运用广义线性量子变换理论和定态微扰论对线型多原子分子振动进行了普遍求解,将线型多原子分子简谐振动能量本征值和态的求解转化成正定和半正定矩阵的对角化问题,微扰矩阵元、能量和波函数各级修正的计算转换到多维无耦合谐振子的本征表象中进行.并以CO2分子为例进行了具体求解.     

Application of Huang-Yang pseudopotential to a few-body system

Huang and Yang replace a hard core of radius c by their pseudopotential (HYPP), which can be used in perturbation theory. I show that HYPP and boundary value perturbation theory give the same energy shift, to first order in c. I model a two-body potential as an oscillator potential (angular frequency ω) and a hard core. For an A-body system, the ratio of the first order energy correction to the unperturbed energy is 0.448 A^3/4 c a, where a = (M ω/2h)^1/2.     

Application of the electrostatic mean value theorem to electrostatic sensor electrodes

We present a method for approximating the potential of conducting objects due to a known electrostatic source. The method involves averaging the incident potential over the conductor surface or volume region, which is known to give the exact value for a perfectly conducting sphere. The method is extended to spheroidal geometries, both prolate and oblate, to study the error incurred for deviations from the spherical case. Exact values for the spheroid potentials are derived and compared with those obtained by the mean value approximations. The result for the oblate spheroid is extended to the case of a two-dimensional electrostatic disk. The approximations are proposed as a method for predicting the potential of conducting electrodes used with electrostatic sensors for the measurement of electrostatic field disturbances. In this regard, the mean value approximation is applied to determine the source to electrode mutual capacitance, which is implemented in the model for the sensor system. Electrostatic disk electrodes are used with an electrostatic disturbance sensor to experimentally validate the application of the mean value approximation.     

Quantitative analysis of Fermi resonances by harmonic derivatives of perturbation theory corrections

Vibrational perturbation theory has proven to be a highly accurate and efficient method for extending the harmonic approximation in the treatment of polyatomic molecular vibrations. Unfortunately, accidental near-degeneracies of the harmonic vibrational levels can lead to resonance and a breakdown of the perturbation approximation. These resonances can be resolved by the diagonalization of a small effective Hamiltonian derived from either of the usual Rayleigh–Schrödinger or van Vleck perturbation theories. However, the proper choice of states for inclusion in the effective Hamiltonian is crucial to the accuracy of the results, and is not often clearly evident. It is proposed that the analytical partial derivatives of the anharmonic vibrational correction with respect to the various harmonic frequencies, called ‘Harmonic Derivatives’ in this work, can be used as a tool to quantitatively assess the existence and strength of first-order, or Fermi, resonances. These derivatives are shown to concisely and clearly reflect the quality of the perturbation approximation and the effect of its breakdown on the computed vibrational levels.     

A Class of Representations of the ∗-Algebra of the Canonical Commutation Relations over a Hilbert Space and Instability of Embedded Eigenvalues in Quantum Field Models

Abstract

In models of a quantum harmonic oscillator coupled to a quantum field with a quadratic interaction, embedded eigenvalues of the unperturbed system may be unstable under the perturbation given by the interaction of the oscillator with the quantum field. A general mathematical structure underlying this phenomenon is clarified in terms of a class of Fock space representations of the ?-algebra of the canonical commutation relations over a Hilbert space. It is also shown that each of the representations is given as a composition of a proper Bogolyubov (canonical) transformation and a partial isometry on the Fock space of the representation.     

Two-site model and its relation to the polaron-crystal model

A symmetric two-site, one-electron model is treated within perturbation theory in electron wavefunction overlap from different sites. The significant role of the double degeneracy of levels in the unperturbed Hamiltonian is pointed out. It is shown that as a result of this the terms of the perturbative expansion for the correction to the energy of an unperturbed level produce different results depending on the parity of the order. Namely, even terms give rise to a level shift (an analog of the polaron shift in the polaron-crystal model), while odd terms result in splitting (an analog of the polaron band width). Also, the shift of the levels decreases with increasing vibronic coupling constant by a power law, and their splitting, exponentially. This is in full accord with the well-known results of small-radius polaron theory. Fiz. Tverd. Tela (St. Petersburg) 39, 2159–2167 (December 1997)     

Electronic states of on- and off-center donors in quantum rings of finite width

The electronic states of a hydrogenic donor in two-dimensional quantum rings are calculated by taking into account the finite width of the potential well in the ring. In addition, a strong magnetic field is applied perpendicular to the quantum ring. Using the effective-mass approximation at the Γ valley, the radial Hamiltonian for the envelope-function is exactly diagonalized in the case of on-center donors. The corresponding energy levels for different angular momenta are studied as a function of the applied magnetic field. In the case of off-center donors, a perturbation approach is considered and its limitations are discussed. Finally, we calculate the absorption spectra and oscillator strength for different intraband transitions, specifically for on-center donors.     

New Variational Perturbation Theory Based on q−Deformed Oscillator

A new variational perturbation theory is developed based on the q−deformed oscillator. It is shown that the new variational perturbation method provides 200 and 10 times better accuracy for the ground state energy of anharmonic oscillator than the Gaussian and the post Gaussian approximation, respectively, for weak coupling.     

Deformed quantum harmonic oscillator with diffusion and dissipation

A master equation for the deformed quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived in the microscopic model by using perturbation theory. The coefficients of the master equation and of equations of motion for observables depend on the deformation function. The steady-state solution of the equation for the density matrix in the number representation is obtained and the equilibrium energy of the deformed harmonic oscillator is calculated in the approximation of small deformation.     

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.     

Effects of vibrational anharmonicity on 19F nuclear resonance in SF6

A Barker-Henderson like perturbation theory for polyatomic fluids is developed. The molecular interaction forces are assumed to be described by an interaction site model potential and the reference system is a fluid of hard interaction site model molecules.

The theory is used to study the equation of state of nitrogen, the theoretical results being compared with experimental data and with those coming from other theories. The agreement between theory and experiment is as good as that shown by Barker and Henderson theory for monoatomic systems.     

Modeling multilayer adsorption of interacting polyatomic species on heterogeneous surfaces

In the present work, a generalized lattice-gas model to study multilayer adsorption of interacting polyatomic species on heterogeneous surfaces is introduced. Using an approximation in the spirit of the well-known Brunauer-Emmet-Teller (BET) model, a new theoretical isotherm is obtained in one- and two-dimensional lattices and compared with Monte Carlo simulation. In addition, the BET approach is used to analyze these isotherms and to estimate the monolayer volume. In all cases, the application of the BET equation leads to an underestimate of the true monolayer capacity. However, significant compensation effects were observed for heterogeneous surfaces and attractive lateral interactions.     

Axially symmetrical molecules in electric and magnetic fields: energy spectrum and selection rules

In this paper we investigate the effects of external electric and magnetic fields on a three-dimensional harmonic oscillator with axial symmetry. The energy spectrum of such a system is non-degenerate due to the presence of the magnetic field. The degeneracy of the energy spectrum in the absence of a magnetic field is discussed. The influence of electric and magnetic fields, as well as the frequencies of the oscillator on the probability distribution function is analyzed. Optical transition probabilities are examined by deriving the selection rules in dipole approximation for the quantum numbers n _p , m _l and n _z . Employing stationary perturbation theory, the effects of deformations of the potential energy function on the oscillatory states are analyzed. Such models have been used in literature in analysis of spectra of axially symmetrical molecules and cylindrical quantum dots.     

MLIMCL：基于机器学习的隐式溶剂蒙特卡罗方法

对于分子结构的优化和预测,蒙特卡罗(MC)是很重要的计算工具. 当溶剂效应被显式的考虑时,由于水分子和电离子的自由度很大,蒙特卡罗方法变得非常昂贵. 相对而言,基于隐式溶剂的蒙特卡罗方法可以通过对溶剂效应平均场的近似来大大降低计算成本,同时还能保持目标分子在原子水平上的细节. 目前两种最流行的隐式溶剂模型是泊松-波兹曼模型和通用化波恩模型. 通用化波恩模型是泊松-波兹曼模型的近似,但在模拟计算时间上要快得多. 本文通过结合两种隐式溶剂模型在准确性和效率方面的优势,开发了一种基于机器学习的隐式溶剂蒙特卡罗方法. 具体而言,蒙特卡罗方法通过机器学习既保留了泊松-波兹曼模型的精度,又达到了通用化波恩模型的速度,从而能快速准确地获取模拟计算中每一步的静电溶解自由能. 本文采用苯-水系统和蛋白质-水系统来验证我们的蒙特卡罗方法. 实验证明蒙特卡罗方法在分子结构优化和预测的速度和准确性方面具有很大优势.     

双原子分子非谐振转波函数和能级

从双原子分子简谐势近似波函数出发,运用微扰理论计算出了双原子分子在非谐振转相互作用下的一级、二级近似能级和一级近似波函数.