Splitting of Spectra in Anharmonic Oscillators Described by Kratzer Potential Function

A perturbation theory model that describes splitting of the spectra in highly symmetrical molecular species in electrostatic field is proposed. An anahrmonie model of a two-dimensional oscillator having Kratzer potential energy function is used to model the molecular species and to represent the unperturbed system. A selection rule for the radial quantum number of the oscillator is derived. The eigenfunctions of a two-dimensional anharmonic oscillator in cylindrical coordinates are used for the matrix elements representing the probability for energy transitions in dipole approximation to be calculated. Several forms of perturbation operators are proposed to model the interaction between the polyatomic molecular species and an electrostatic field. It is found that the degeneracy is removed in the presence of the electric field and spectral splitting occurs. Anharmonic approximation for the unperturbed system is more accurate and reliable representation of a reaJ polyatomic molecular species.     

Nonsensitive Homotopy-Pade Approach： Anharmonic Oscillators

In order to investigate a complicated physical system, it is convenient to consider a simple, easy to solve model, which is chosen to reflect as much physics as possible of the original system, as an ideal approximation. Motivated by this fundamental idea, we propose a novel asymptotic method, the nonsensitive homotopy-Pade approach. In this method, homotopy relations are constructed to link the original system with an ideal, solvable model. An artificial homotopy parameter is introduced to the homotopy relations as the normal perturbation parameter to generate the perturbation series, and is used to implement the Padd approximation. Meanwhile, some other auxiliary nonperturbative parameters, which are used to control the convergence of the perturbation series, are inserted to the approximants, and are fixed via the principle of minimal sensitivity. The method is used to study the eigenvalue problem of the quantum anharmonic oscillators. Highly accurate numerical results show its validity. Possible further studies on this method are also briefly discussed.     

Quasiaccurately solvable quantum mechanics problems and the anharmonic oscillator problem

The basic methods of obtaining the anharmonic oscillator spectrum are briefly reviewed. An approach is discussed to this problem from the point of view of one of the quasiaccurately solvable models of quantum mechanics, whose wave functions are applicable to the construction of a regular perturbation theory. Results are presented of numerical calculations of the ground state energy. Analytic calculations are provided of integrals of the form , occurring in applications of the method of quasiaccurately solvable problems. M. V. Lomonosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 76–88, February, 1993.     

Numerical experiment of anharmonic oscillators by using the symplectic scheme-shooting method

Symplectic scheme-shooting method (SSSM) is applied to solve the energy eigenvalues of anharmonic oscillators characterized by the potentials V(x)=λx4 and V(x)=(1/2)x2+λx2α with α=2,3,4 and doubly anharmonic oscillators characterized by the potentials V(x)=(1/2)x2+λ1x4+λ2x6, and a high order symplectic scheme tailored to the "time"-dependent Hamiltonian function is presented. The numerical results illustrate that the energy eigenvalues of anharmonic oscillators with the symplectic scheme-shooting method are in good agreement with the numerical accurate ones obtained from the non-perturbative method by using an appropriately scaled basis for the expansion of each eigenfunction; and the energy eigenvalues of doubly anharmonic oscillators with the sympolectic scheme-shooting method are in good agreement with the exact ones and are better than the results obtained from the four-term asymptotic series. Therefore, the symplectic scheme-shooting method, which is very simple and is easy to grasp, is a good numerical algorithm.     

Step Operators for the Quasi-exactly Solvable Systems

In this paper, we investigate the step operators for the quasi-exactly solvable problems. We also discuss the commutation relations between the step operators and the Hamiltonian of the quasi-exactly solvable system. After obtaining the general results, we take the anharmonic oscillators with x ⁶ anharmonicity in quasi-exactly solvable problems as examples to give the specific forms of step operators.     

Alternative perturbation expansions for partially solvable quantum-mechanical models

We discuss alternative perturbation expansions for partially solvable quantum-mechanical models. We consider a sextic anharmonic oscillator as a particular example.     

Critical evaluation of anharmonic corrections to the equilibrium isotope effect for methyl cation transfer from vacuum to dielectric continuum

Harmonic and anharmonic vibrational frequencies are computed for isotopologues of methyl cation in vacuum and in a polarised continuum model (PCM) dielectric continuum (? = 80) within Gaussian09. Comparison of results in vacuum for two methods (B3LYP and second-order Møller–Plesset perturbation theory) and three basis sets (6-31+G(d), cc-aug-PVDZ, cc-aug-PVQZ) with published anharmonic frequencies obtained from an accurate vibrational configuration interaction (VCI) method shows the smallest root mean square error in the frequencies from B3LYP/6-31+G(d) with anharmonic corrections. Using this method to calculate isotopic partition function ratios (IPFRs) for all six pairs of CH₃⁺, CH₂D⁺, CHD₂⁺ and CD₃⁺ gives better results for anharmonic frequencies than for unscaled harmonic frequencies, but scaled harmonic frequencies give even better results for less cost. The scaling factor is simply the ratio of the sum of the anharmonic VCI frequencies to the sum of the harmonic B3LYP/6-31+G(d) frequencies, which corresponds to the dominance of zero-point energy changes in determining the IPFRs. Both the scaled and unscaled harmonic frequencies provide reasonable estimates for the equilibrium isotope effects (EIEs) upon transfer of methyl cation from vacuum to PCM ‘water’, but the anharmonic PCM calculations give erratic results. The use of scaled B3LYP/6-31+G(d) harmonic frequencies is recommended for the estimations of EIEs rather than expensive anharmonic corrections.     

Calculation of anharmonic effects in the unimolecular dissociation of M2+(H2O)2 (M = Be,Mg, and Ca)

The anharmonic and harmonic rate constants of the unimolecular dissociation of M²⁺(H₂O)₂ (M = Be, Mg, and Ca) were calculated using the Rice–Ramsperger–Kassel–Marcus theory. The anharmonic effects of the reactions were investigated. The results show that the energy barrier of the dissociation of Be²⁺(H₂O)₂ is 68.47 kcal/mol, and the anharmonic (T_4000K = 4.28×10⁸ s^?1) and harmonic (T_4000K = 4.22×10⁸ s^?1) rate constants were close in value in both the canonical and microcanonical systems. The energy barriers of the two steps for the dissociation, Mg²⁺(H₂O)₂ → MgOH⁺+H₃O⁺, were 37.41 and 11.39 kcal/mol, and those for the dissociation, Ca²⁺(H₂O)₂ → CaOH⁺+H₃O⁺, were 21.15 and 26.42 kcal/mol. The anharmonic effect of the two reactions is significant and cannot be neglected in both the canonical and microcanonical systems. The comparison also shows that the rate constants of the dissociation of Ca²⁺(H₂O)₂ have the maximum values, while those of Be²⁺(H₂O)₂ have the minimum values in the three reactions; however, the anharmonic effect also shows the similar trend in the comparison.     

A New Quasi-Exactly Solvable Problem and Its Connection with an AnharmonicOscillator

The two-dimensional hydrogen with a linear potential in a magnetic field is solved by two different methods. Furthermore the connection between the model and an anharmonic oscillator is investigated by methods of KS transformation.     

Exactly solvable model of a structural phase transition

An exactly solvable d-dimensional model for the structural phase transition with long-range anharmonic interaction is considered. Classical as well as purely quantum regimes are discussed within the framework of the method of approximating Hamiltonians.     

Dynamic renormalization-group analysis of the d+1 dimensional Kuramoto-Sivashinsky equation with both conservative and nonconservative noises

The long-wavelength properties of the (d + 1)-dimensional Kuramoto-Sivashinsky (KS) equation with both conservative and nonconservative noises are investigated by use of the dynamic renormalization-group (DRG) theory. The dynamic exponent z and roughness exponent α are calculated for substrate dimensions d = 1 and d = 2, respectively. In the case of d = 1, we arrive at the critical exponents z = 1.5 and α = 0.5 , which are consistent with the results obtained by Ueno et al. in the discussion of the same noisy KS equation in 1+1 dimensions [Phys. Rev. E 71, 046138 (2005)] and are believed to be identical with the dynamic scaling of the Kardar-Parisi-Zhang (KPZ) in 1+1 dimensions. In the case of d = 2, we find a fixed point with the dynamic exponents z = 2.866 and α = -0.866 , which show that, as in the 1 + 1 dimensions situation, the existence of the conservative noise in 2 + 1 or higher dimensional KS equation can also lead to new fixed points with different dynamic scaling exponents. In addition, since a higher order approximation is adopted, our calculations in this paper have improved the results obtained previously by Cuerno and Lauritsen [Phys. Rev. E 52, 4853 (1995)] in the DRG analysis of the noisy KS equation, where the conservative noise is not taken into account.     

Lie Algebraic Approach to Energy Transfer for Collinear Collision of Two Anharmonic Diatomic Molecules

An anharmonic oscillator algebra model is used to study the collinear collisions of two diatomic molecules. The transition probability for vibration-vibration energy transfer is presented. For an application of the method, we talk about the collision of N2＋CO, N2＋O2, and N2＋N2. Through long time averaging, the transition probability changes to the function of total energy of the system. Comparing the results with the quantum results, we can see that the dynamical Lie algebraic method is useful for describing the anharmonie diatomic molecular collision.     

Boson factorization approximation for anharmonic nuclear vibrations

A new approach is proposed for treating anharmonic nuclear vibrations, based on a kind of Hartree factorization applied to the Hamiltonian in a boson representation. The method is illustrated with the exactly solvable model of Lipkin, Meshkov and Glick.     

Ising Transition in Dimerized XY Quantum Spin Chain

We proposed a simple spin-1/2 model which provides an exactly solvable example to study the Ising criticality with central charge c=1/2.By mapping it onto the real Majorana fermions,the Ising critical behavior in explored explicitly,although its bosonized form is not the double frequency sine-Gordon model.     

Temperature dependence of liquid volume

It is shown that the temperature dependence of the liquid volume is well described by the equation $ V = A + BT + CT^2 + V_e \exp ( - E/RT), $ V = A + BT + CT^2 + V_e \exp ( - E/RT), where A, B, C, V _e , and E are constants. This equation reflects two processes owing to which the liquid volume increases with temperature, namely, anharmonic oscillations of molecules and formation of “holes.”     

The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit

We study the free path length and the geometric free path length in the model of the periodic two-dimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the small-scatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term in the asymptotic formula of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.In memory of Walter Philipp     

非谐振子势的精确解和双波函数描述

求解了非谐振子势V(x)=x²/2+g/2x²的本征方程，给出了精确的能谱方程和归一化波函数.应用双波函数理论，得到了在非谐振子势场中单粒子运动状态的力学量的时间演化方程. 关键词：     

Quantum Ising Criticality in Dimerized XY Spin-1/2 Chain

In the present paper we propose a spin-1/2 chain which provides an exactly solvable example to studythe Ising criticality with the central charge c = 1/2. We performthe diagonalization of this model in the presence ofmagnetic field. From the full energy spectrum, the central charge and the scaling dimensions are given at the criticalpoint. The results show evidently that the quantum Ising criticality exists in such a system.     

Remarks on interpretations of the Eotvos experiment and misinterpretation of E=mc^2

The Eotvos experiment on the verification of equivalence between inertial mass and gravitational mass of a body is famous for its accuracy. A question is, however, can these experimental results be applied to the case of a physical space in general relativity, where the space coordinates could be arbitrary？ It is pointed out that it can be validly applied because it has been proven that Einstein＇s equivalence principle for a physical space must have a frame of reference with the Euclidean-like structure. Will claimed further that such an overall accuracy can be translated into an accuracy of the equivalence between inertial mass and each type of energy. It is shown that, according to general relativity, such a claim is incorrect. The root of this problem is due to an inadequate understanding of special relativity that produced the famous equation E=mc^2, which must be understood in terms of energy conservation. Concurrently, it is pointed out that this error is a problem in Will＇s book, ‘Theory and Experiment in Gravitational Physics＇.