Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.     

氧化亚氮3000—0200和1001—0110跃迁带在高温下的线强度

采用乘积近似法计算了氧化亚氮分子的总配分函数，其中转动配分函数考虑了离心扭曲修正，振动配分函数采用谐振子近似. 利用计算所得的配分函数和实验振动跃迁矩平方及Herman-Wallis因子系数，计算了氧化亚氮3000—0200和1001—0110跃迁带在常温和高温下的线强度. 结果显示，当温度高达3000K时，计算所得线强度与实验值及HITRAN数据库提供的结果仍符合较好. 这表明高温下的分子配分函数和线强度的计算是可靠的. 还进一步计算了氧化亚氮3000—0200和1001—0110跃迁带在更高温度(40 关键词： 氧化亚氮 配分函数 线强度 高温     

Thermal zeta functions

The partition function for a simple harmonic oscillator and the fermion equivalent is evaluated using a path integral technique based upon generalized zeta functions.     

Study on high-temperature spectra of asymptotic asymmetric-top radical SiO2

Total internal partition sums are calculated in the product approximation at temperatures up to 6000 K for the asymptotic asymmetric-top SiO2 molecule.The rotational partition function and the vibrational partition function are calculated with the rigid-top model and in the harmonic oscillator approximation,respectively.Our values of the total internal partition sums are consistent with the calculated value in the Gaussian program within 0.137% at 296 K.Using the calculated partition functions and the rotationless transition dipole moment squared as a constant,we calculate the line intensities of 001-000 band of SiO2 at normal,medium and high temperatures.Simulated spectra of the 001-000 band of the asymptotic asymmetric-top SiO2 molecule at 2000,5000 and 6000 K are also obtained.     

Quantum vibrational partition function in the non-extensive Tsallis framework

The quantum vibrational partition function has been obtained in the Tsallis statistics framework for the entropic index, q, between 1 and 2. The effect of non-extensivity on the population of states and thermodynamic properties have been studied and compared with their corresponding values obtained in the Boltzmann-Gibbs (BG) statistics. Our results show that the non-extensive partition function of harmonic oscillator at any temperature is larger than its corresponding values for an extensive system and that their differences increase with temperature and entropic index. Also, the number of accessible states increases with q but, compared to the BG statistics, the occupation number decreases for low energy levels while the population of the higher energy levels increases. The internal energy and heat capacity have also been obtained for the non-extensive harmonic oscillator system. Results indicate that the heat capacity is greater than its corresponding value in the extensive (BG) system at low temperatures but that this trend is reversed at higher temperatures.     

类Quesne环状球谐振子势场中的赝自旋对称性

提出了一种新的类Quesne环状球谐振子势,应用二分量方法求解1/2-自旋粒子满足的Dirac方程, Dirac哈密顿量由标量和矢量类Quesne环状球谐振子势构成.在Σ=S(r)+V(r)=0的条件下,得到了Dirac旋量波函数下分量的束缚态解和能谱方程, 显示出类Quesne环状球谐振子势场中的赝自旋对称性.讨论了束缚态波函数和能谱方程的有关性质. 关键词： 类Quesne环状球谐振子势 Dirac方程 赝自旋对称性 束缚态     

非对易相空间中谐振子体系热力学性质的探讨

2000年以来, 有关非对易空间的各种物理问题一直是研究的热点, 并在量子力学、场论、凝聚态物理、天体物理等各领域中已被广泛地探讨. 采用统计物理方法讨论非对易效应对谐振子体系热力学性质的影响. 先以对易相空间中确定二维和三维谐振子的配分函数求出谐振子体系的热力学函数; 非对易相空间中的坐标和动量通过坐标-坐标和动量-动量之间的线性变换而以对易相空间中的坐标和动量来表示; 最终以非对易相空间中求出配分函数来讨论非对易效应对谐振子体系热力学性质的影响. 结果显示, 在非对易相空间中谐振子体系的配分函数和熵表达式均包含因非对易引起的修正项. 从分析结果得出如下结论: 非对易效应对谐振子的配分函数和熵函数等微观状态函数有一定的影响, 但对谐振子体系的内能、热容量等宏观热力学函数没有影响. 研究结果只是对应于满足玻尔兹曼统计的经典体系, 对于满足费米-狄拉克和玻色-爱因斯坦统计的量子体系需进一步推广研究.     

EXACT SOLUTIONS TO THE N-BODY SCHRÖDINGER EQUATION FOR THE HARMONIC OSCILLATOR

The exact solutions to the N-body Schrödinger equation for the harmonic oscillator are presented analytically. The permutational symmetry of the solutions for the identical three-body system of the harmonic oscillator are discussed in some detail.     

Zeta Functions for the Spectrum of the Non-Commutative Harmonic Oscillators

This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in [PW1, 2]. It is shown, as one of the basic analytic properties, that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at s=1, and further that it has a zero at all non-positive even integers, i.e. at s=0 and at those negative even integers where the Riemann zeta function has the so-called trivial zeros. As a by-product of the study, both the upper and the lower bounds are also given for the first eigenvalue of the non-commutative harmonic oscillator.Work in part supported by Grant-in Aid for Scientific Research (B) No. 16340038, Japan Society for the promotion of ScienceWork in part supported by Grant-in Aid for Scientific Research (B) No. 15340012, Japan Society for the promotion of Science     

Asymptotic Lower Bound for the Relative Disparities of Truncated-Path-Integral Partition Functions

Any truncated-path-integral partition function of a nonrelativistic quantum system in thermodynamic equilibrium—one obtained by means of the Feynman path-integral-procedure using a finite number of such integrals—is known to have a value not less than that of the exact one corresponding to it. A rigorous asymptotic lower bound obtained for the relative disparity in their values—the difference in their values divided by that of the exact partition function— confirms asymptotic positive-definiteness of the original upper bound. Values determined directly for a linear harmonic oscillator agree asymptotically with values of they bound.     

Wave function of the harmonic oscillator in classical statistical mechanics

The notion of wave function of the classical harmonic oscillator is discussed. The evolution equation for this wave function is obtained using the classical Liouville equation for the probability-distribution function of the harmonic oscillator. The tomographic-probability distribution of the classical oscillator is studied. Examples of the ground-like state and the coherent state of the classical harmonic oscillator are considered.     

关于谐振子第一积分的研究

提出一维谐振子基本积分的概念, 并利用它们构造其他积分. 将上述概念和方法推广到多维谐振子, 利用直接构造法证明不同类型的二维谐振子都存在三个与时间无关的独立的第一积分, n维谐振子系统存在2n-1个与时间无关的独立的第一积分. 讨论了分振动频率比为有理数和无理数的二维非对称谐振子的特征. 关键词： 谐振子 第一积分 基本积分 不含时积分     

Steady-State Responses of NRD-Guide LSE/LSM Mode Oscillators in MM-Wave Bands

The analysis methods for the steady-state responses of the mm-wave band NRD-guide negative impedance oscillators based on nonlinear microwave autonomous circuits harmonic balanced method are presented in the paper in details. Firstly, the large-signal nonlinear lumped equivalent circuits of the Gunn diode are studied in mm-wave bands. Then, the performances of two kinds of NRD-guide autonomous circuits, the probe-exciting LSM ₁₁ ^o-mode oscillator and the LSE ₁₁ ^o-mode oscillator, are analyzed by the way of extracting the large-signal dynamic harmonic admittance (conductance) of the Gunn diode or directly configuring the harmonic balance equations for the oscillator. The input impedance of the exciting probe in the oscillator and the performances of the load-pulling and the local stability of the NRD-guide oscillator are also involved.     

虚时路径积分的广义四阶拆分方法及其谐振子分析

虚时路径积分方法是计算实际体系量子热力学性质的有效工具. 本文重点分析讨论了基于二阶和四阶拆分的虚时路径积分的有效性,给出谐振子体系配分函数统一的解析形式,进而得到其热力学性质和主要误差的表达式. 通过去除谐振子配分函数的主要误差,广义对称四阶拆分中的自由参数得以固定,以此计算实际体系的热力学量可以达到期望的三阶精度.     

Semileptonic (Λ b → Λcev) decay in a field theoretic quark model

The semileptonic decay width of heavy baryons such as (Λ _b → Λcev) has been estimated in the framework of a nonrelativistic field theoretic quark model where four component quark field operators along with a harmonic oscillator wave function are used to describe translationally invariant hadronic states. The present estimation does not make an explicit use of heavy quark symmetry and has a reasonable agreement with the experimentally measured decay width, polarisation ratio and form factors with the harmonic oscillator radii and quark momentum distribution inside the hadron as free parameters.     

The quantum harmonic oscillator on a circle and a deformed Heisenberg algebra

We show that the Heisenberg-type algebra describing the first levels of the quantum harmonic oscillator on a circle of large length L is a deformed Heisenberg algebra. The successive energy levels of this quantum harmonic oscillator on a circle of large length L are interpreted, similarly to the standard quantum one-dimensional harmonic oscillator on an infinite line, as being obtained by the creation of a quantum particle of frequency w at very high energies. Received: 29 March 2001 / Revised version: 17 July 2001 / Published online: 31 August 2001     

Line intensities of the asymptotic asymmetric-top radical HO2 at high temperatures

The total internal partition sums were calculated in the product approximation at temperatures up to 5000\,K for the asymptotic asymmetric-top HO$_{2}$ molecule. The calculations of the rotational partition function and the vibrational partition function were carried out with the rigid-top model and in the harmonic oscillator approximation, respectively. Our values of the total internal partition sums are consistent with the data of HITRAN database with $-$0.14{\%} at 296\,K. Using the calculated partition functions, we have calculated the line intensities of $\nu _{2}$ band of HO$_{2}$ at several high temperatures. The results showed that the calculated line intensities are in very good agreement with those of HITRAN database at temperatures up to 3000\,K, which provides a strong support for the calculations of partition functions and line intensities at high temperatures. Then we have extended the calculation to higher temperatures. The simulated spectra of $\nu_{2}$ band of the asymptotic asymmetric-top HO$_{2}$ molecule at 4000 and 5000\,K are also obtained.     

Time Dependence of Joint Entropy of Oscillating Quantum Systems

The time dependent entropy (or Leipnik’s entropy) of harmonic and damped harmonic oscillator systems is studied by using time dependent wave function obtained by the Feynman path integral method. The Leipnik entropy and its envelope change as a function of time, angular frequency and damping factor. Our results for simple harmonic oscillator are in agreement with the literature. However, the joint entropy of damped harmonic oscillator shows remarkable discontinuity with time for certain values of damping factor. The envelope of the joint entropy curve increases with time monotonically. These results show the general properties of the envelope of the joint entropy curve for quantum systems.     

Spectral inverse problem for q-deformed harmonic oscillator

The supersymmetric quantization condition is used to study the wave functions of SWKB equivalent q-deformed harmonic oscillator which are obtained by using only the knowledge of bound-state spectra of q-deformed harmonic oscillator. We have also studied the nonuniqueness of the obtained interactions by this spectral inverse method.     

A Relativistic Model of the Isotropic Oscillator

For a three-dimensional model of the harmonic oscillator in the relativistic configurational r-representation wave functions in the spherical coordinates r = (r, Θ, ?) are found. The generating function, orthogonality and various recurrence relations for the radial part of the wave function are obtained. The radial and orbital quantum numbers raising and lowering operators are defined and the dynamical symmetry group is constructed by means of the Infeld-Hull factorization method.