A new approach to obtain the non-Condon factors in closed form for two one-dimensional harmonic oscillators

A simple algebraic approach to calculate general Franck-Condon overlaps is extended to evaluate non-Condon factors for two one-dimensional harmonic oscillators. The method is based on the use of eigenstates of the harmonic oscillator annihilation operator which allows to obtain in terms of a multi-dimensional Hermite polynomial the overlap of harmonic oscillator functions associated with different Born-Oppenheimer potentials. The presented approach is self-contained, only basic concepts of quantum mechanics associated with the harmonic oscillator system are needed. The obtained expression for the Franck-Condon overlaps is similar to the Ansbacher’s formula and equivalent to the one calculated by Malkin and Man’ko. However our final expression has the advantages that only real numbers are involved and it is straightforward to get the limit case of equal frequencies. Concerning the non-Condon factors two approaches leading to different formulas are considered, both of which reduce to triple sums of products of three Hermite polynomials.     

The calculating formula for radial matrix elements of a relativistic harmonic oscillator

A universal practical formula is given for calculating an integral which includes two confluent hypergeometric functions, power and exponential functions; then by means of this formula, the expressions of the radial matrix elements for a relativistic harmonic oscillator are given.     

Extended Feynman formula for time-dependent forced harmonic oscillator

Horváthy's modification of Feynman's path integral formula is generalized to the time-dependent forced harmonic oscillator. The propagator at caustics is then obtained by using its modified semi-group property. Finally, with our new formula, the propagator for a charged particle in a time-dependent electromagnetic field is evaluated exactly beyond and at caustics.     

Path-integral expression of dissipative dynamics

The Feynman-Ito formula for a wide class of bounded absorptive potentials as well as for the multidimensional damped harmonic oscillator is derived within the framework of the path-integral theory of Albeverio, Høegh-Krohn and Truman.     

Remark on the solution of the Schrödinger equation for anharmonic oscillators via the Feynman path integral

We give a simple proof of Feynman's formula for the Green's function of the n-dimensional harmonic oscillator valid for every time t with Im t ≤ 0. As a consequence the Schrödinger equation for the anharmonic oscillator is integrated and expressed by the Feynman path integral on Hilbert space.     

Physical principles in quantum field theory and in covariant harmonic oscillator formalism

It is shown that both covariant harmonic oscillator formalism and quantum field theory are based on common physical principles which include Poincaré covariance, Heisenberg's space-momentum uncertainty relation, and Dirac's “C-number” time-energy uncertainty relation. It is shown in particular that the oscillator wave functions are derivable from the physical principles which are used in the derivation of the Klein-Nishina formula.     

On the off-diagonal operator equivalents

A closed polynomial formula for the qth component of any off-diagonal operator equivalent of order k (integer or half-integer) is derived in terms of two-dimensional harmonic oscillator creation and annihilation operators.     

Feynman's formula for a harmonic oscillator

Feynman's original formula for the Green's function of a harmonic oscillator includes the Maslov correction. It also leads to the known formula for caustics for the time intervals that are equal to integer multiplets of the half period.     

分数阶质量涨落谐振子的共振行为

针对黏性介质引起的Brown粒子质量存在随机涨落以及阻尼力对历史速度具有记忆性等问题, 本文首次提出分数阶质量涨落谐振子模型, 以考察黏性介质中Brown粒子的动力学特性. 首先, 将Shapiro-Loginov 公式分数阶化, 使之适用于对含指数关联随机系数的分数阶随机微分方程的求解. 然后, 利用随机平均法和分数阶Shapiro-Loginov公式推导系统稳态响应振幅的解析表达式, 并据此研究系统的共振行为; 最后, 通过仿真实验验证理论结果的可靠性. 研究表明: 1)质量涨落噪声可诱导系统产生随机共振行为; 2)记忆性阻尼力可诱导系统产生参数诱导共振行为; 3)不同参数条件下, 系统表现出单峰共振、双峰共振等多样化的共振形式. 关键词： 黏性介质 质量涨落 阻尼记忆性 分数阶谐振子     

Extended Feynman formula for harmonic oscillator

A slight modification of Feynman's original method leads to the Maslov correction in the path integral formula of a harmonic oscillator. Caustics are treated in a direct geometric way.On leave from Veszprém University of Chemical Engineering (Hungary).     

N维各向同性谐振子径向矩阵元的通项表达式

推导出了N维(N≥2)各向同性谐振子径向矩阵元的通项表达式,所得结果包含了以前人们给出的N=2、3时的结果。     

New Poisson Sum Formula Derived by Squeezing Transformation in Quantum Optics

Using squeezing transform in the context of quantum optics and based on the Fourier series expansion we rigorously derive a new Poisson sum formula. Application of this new formula to the representation transformation of kq-wave function for describing electrons in periodic lattice is demonstrated. In so doing, the transition matrix element of harmonic oscillator in kq representation is derived.     

Oscillator parameters in nuclei

The dependence on nucleon numbers of the harmonic oscillator length parameter b or energy spacing ℏω in nuclei is determined, using an extensive tabulation of nuclear charge radii and an empirical expression for the difference between proton and neutron radii. Various possible constraints on the parameters or radii are discussed. Alternative parametrizations are compared and a preferred “universal” parametrization is proposed. It is argued that the established Blomqvist–Molinari formula is perfectly adequate for determining the oscillator parameter for specific applications.     

Two-dimensional friction oscillator: group-preserving scheme and handy formulae

With vibration isolation of buildings and turbomachinery blades in mind, we study the dynamic behaviour of a single-mass two-degree-of-freedom oscillator with dry friction damper, viscous damper and elastic spring connected in parallel. The mass is mounted on an elastic supporting plate allowing movement in two directions on a plane. We formulate a multi-dimensional friction model, from which the sliding conditions and the sticking conditions of the mass are derived. For calculations we develop a group-preserving scheme, which preserves the projective proper orthochronous Lorentz group PSO_o(2,1) symmetry of the model in the sliding phase so as to satisfy automatically the sliding conditions at each time step without iteration at all. The oscillator is then subjected to simple harmonic excitations, and the responses are displayed. According to the simple harmonic balance method together with a circular orbit assumption on displacements, we derive closed-form formulae for handily estimating the steady state responses, which are then compared with the results calculated by the group-preserving scheme to confirm the applicability of the formulae. We also derive formulae specifically for a two-dimensional friction oscillator with rigid base support, which include an exact formula of the magnification factor and a simple formula for estimating the minimum driving force amplitude (or the maximum friction force bound) to avoid sticking.     

磁绝缘线振荡器中空间电荷的辐射

 研究了磁绝缘线振荡器中空间电荷的辐射。结果包括：导出了线性区的增益公式，所得到的增益函数相对于同步失配是对称的；导出了饱和区的辐射场公式，这个公式表明，饱和辐射功率近似地与谐波电流的平方成正比。这些理论结果部分地由数值模拟证实。     

二维各向同性谐振子径向矩阵元的递推关系

推导出二维各向同性谐振子径向矩阵元〈nm|ρ^q|n′m′〉所满足的递推关系.     

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     

On the Wigner distribution function for an oscillator

We present two new derivations of the Wigner distribution function for a simple harmonic oscillator Hamiltonian. Both methods are facilitated using a formula which expresses the Wigner function as a simple trace. The first method of derivation then utilizes a modification of a theorem due to Messiah. An alternative procedure makes use of the coherent state representation of an oscillator. The Wigner distribution function gives a semiclassical joint probability for finding the system with given coordinates and momenta, and the joint probability is factorable for the special case of an oscillator. An important application of this result occurs in the theory of nuclear fission for calculating the probability distributions for the masses, kinetic energies, and vibrational energies of the fission fragments at infinite separation.     

均匀磁场中三维各向同性谐振子微扰矩阵元的普遍表达式

根据Laguerre多项式的积分公式,在考虑了哈密顿量中正比于B~2项影响的情况下,导出了均匀磁场中三维各向同性谐振子微扰矩阵元的普遍表达式,并指出n=3能级的简并度完全解除．     

A formula for charmonium suppression

In this work a formula for charmonium suppression obtained by Matsui in 1989 is analytically generalized for the case of complex $c\bar c$ potential described by a 3-dimensional and isotropic time-dependent harmonic oscillator (THO). It is suggested that under certain scheme the formula can be applied to describe J/?? suppression in heavy-ion collisions at CERN-SPS, RHIC, and LHC with the advantage of analytical tractability.