Forced Time-Dependent Harmonic Oscillator in a Static Magnetic Field: Exact Quantum and Classical Solutions

Exact wave functions of the forced time-dependent two-dimensional harmonic oscillator in a static magnetic field are derived by unitary transformation. The geometrical phase induced by the driving force is the phase of the de Broglie wave associated with the particle moving according to the classical equation. Extending the idea of the Heisenberg correspondence principle to the time-dependent system, the exact classical solution obtained from quantum matrix elements.     

Even and Odd Coherent States for Time-Dependent Harmonic Oscillator

The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.     

Gauge Invariance of a Time-Dependent Harmonic Oscillator in Magnetic Dipole Approximation

A manifestly gauge-invariant formulation of non-relativistic quantum mechanics is applied to the case of time-dependent harmonic oscillator in the magnetic dipole approximation. A general equation for obtaining gauge-invariant transition probability amplitudes is derived.     

Theoretical Study of Millimeter Wave Harmonic Oscillator Stabilized with a Transmission Cavity

An equivalent circuit model of millimeter wave second harmonic oscillator stabilized with a transmission cavity has been proposed for constructing analytical formulations between performance parameters of the oscillator and parameters of the circuit. The model consists of an equivalent circuit of fundamental wave and that of second harmonic wave. Each of the circuits comprises circuit models of main cavity, transmission waveguide, and transmission cavity. Absorbing material placed between the transmission waveguide and the transmission cavity can suppress additional resonances originated from transmission cavity. The behavior of the second harmonic oscillator can be effectively described by the circuit model. Furthermore, based on this model, mechanical tuning characteristics have been studied at first, and then analytical formulas for quality factor and efficiency depending on circuit parameters have been derived. The circuit parameters can be conveniently extracted by electromagnetic field simulation. Hence the formulas exhibit both compact form and enough accuracy. Thereafter, general rules of performance parameters varying with circuit parameters have been deduced for the harmonic oscillators. Then some design considerations have been derived according to the corresponding analysis. The equivalent circuit model is useful for designing and adjusting millimeter wave second harmonic stabilizing oscillator with a transmission cavity.     

二维非谐振子的束缚态解

讨论势函数为V(ρ)=2ρ λρα的非谐振子的Schr dinger方程。当α=2时其有Hamiltonian量的精确束缚态解。当α3时其有Hamiltonian量的微扰解,尤其是它的一阶、两阶的能量修正及零级、一级波函数。     

Pseudospin Symmetry for a Ring-Shaped Non-spherical Harmonic Oscillator Potential

The pseudospin symmetry for a ring-shaped non-spherical harmonic oscillator potential is investigated by solving the Dirac equation with equal mixture of scalar and vector potentials with opposite signs. The normalized spinor wave function and energy equation are obtained, the algebraic property of the energy equation and some particular cases are also discussed.     

Non-compact Groups,Coherent States,Relativistic Wave Equations and the Harmonic Oscillator

Relativistic geometrical action for a quantum particle in the superspace is analyzed from theoretical group point of view. To this end an alternative technique of quantization outlined by the authors in a previous work and, that is, based in the correct interpretation of the square root Hamiltonian, is used. The obtained spectrum of physical states and the Fock construction consist of Squeezed States (SS) which correspond to the representations with the lowest weights and with four possible (non-trivial) fractional representations for the group decomposition of the spin structure. From the theory of semi-groups the analytical representation of the radical operator in the superspace is constructed, the conserved currents are computed and a new relativistic wave equation is proposed and explicitly solved for the time dependent case. The relation with the Relativistic Schr?dinger equation and the Time-dependent Harmonic Oscillator (TDHO) is analyzed and discussed. An erratum to this article can be found at     

Anisotropic Harmonic Oscillator in a Static Electromagnetic Field

A nonrelativistic charged particle moving in an anisotropic harmonic oscillator potential plus a homogeneousstatic electromagnetic field is studied. Several configurations of the electromagnetic field are considered. The Schrodingerequation is solved analytically in most of the cases. The energy levels and wave functions are obtained explicitly. Insome of the cases, the ground state obtained is not a minimum wave packet, though it is of the Gaussian type. Coherentand squeezed states and their time evolution axe discussed in detail.     

Exact Harmonic Metric for a Uniformly Moving Schwarzschild Black Hole

The harmonic metric for Schwarzschild black hole with a uniform velocity is presented. In the limit of weak field and low velocity, this metric reduces to the post-Newtonian approximation for one moving point mass. As an application, we derive the dynamics of particle and photon in the weak-field limit for the moving Schwarzschild black hole with an arbitrary velocity. It is found that the relativistic motion of gravitational source can induce an additional centripetal force on the test particle, which may be comparable to or even larger than the conventional Newtonian gravitational force.     

Generalized Harmonic Oscillator and the SchrSdinger Equation with Position-Dependent Mass

We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties corresponding effective potentials for several mass functions, for the system with PDM are also discussed. We give the the systems with such potentials are isospectral to the usual harmonic oscillator.     

Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian. PACS: 03.20.+i, 03.30.+p, 03.65.−w,03.65.Ca     

Interpretation of the Quantum Measurement: Example of a Linearly Coupled Harmonic Oscillator

We argue that it may be possible to consistently explain the quantum measurement by assuming that the wave function is in one-to-one correspondence with objective physical reality and has no probabilistic interpretation. In the context of such approach we consider the model of a harmonic oscillator linearly coupled to a heat bath and treat the oscillator as the system being measured. Three classes of initial pure states for the bath are considered. Exact expressions for the average values and variances of the oscillator coordinate and momentum as functions of time are considered for each class of pure states. It is shown that these quantities exhibit different asymptotic behavior for different classes of initial states of the bath. In particular, if each mode of the bath is initially in a coherent state, then for an arbitrary initial state of the oscillator the variances of the oscillator coordinate and momentum asymptotically approach the same values as for a coherent state of the free oscillator, while the averages of coordinate and momentum show a Brownian-like behavior. We argue that such behavior shows several features of the quantum measurement and supports our interpretation of the wave function.     

Normally-Ordered Time Evolution Operator for Mass-Varying Harmonic Oscillator and Wigner Function of Squeezed Number State

For investigating dynamic evolution of a mass-varying harmonic oscillator we constitute a ket-bra integration operator in coherent state representation and then perform this integral by virtue of the technique of integration within an ordered product of operators. The normally ordered time evolution operator is thus obtained. We then derive the Wigner function of u（t）｜n,〉, where ｜n〉 is a Fock state, which exhibits a generalized squeezing, the squeezing effect is related to the varying mass with time.     

Solutions of the Schrodinger Equation with Quantum Mechanical Gravitational Potential Plus Harmonic Oscillator Potential

The solutions of the Schrodinger equation with quantum mechanical gravitational potential plus harmonic oscillator potential have been presented using the parametric Nikiforov-Uvarov method. The bound state energy eigen values and the corresponding un-normalized eigen functions are obtained in terms of Laguerre polynomials. Also a special case of the potential has been considered and its energy eigen values are obtained.     

幂级数展开法计算周期阻尼振子路径积分短时传播子

文章用Schwinger方法与Makri等人发展的幂级数展开法研究了周期阻尼简谐振子的传播子.研究表明,这两种方法所获得的短时传播子非常接近,但是后者比前者要简单得多,这给研究复杂系统的传播子提供了一个简单可靠的方法.     

Calculation of Gaussian Perturbation on Two Harmonic Oscillators by Virtue of Entangled State Representation

The exact expressions of Gaussian-perturbation matrix elements in one- and two-mode Fock states are derived by virtue of the technique of integration within an ordered product of operators and the entangled state representation. It turns out that the matrix elements are just related to Gegenbauer polynomial and Hypergeometric function respectively. The 1st- and 2nd-order corrections to the energy levels and the 1st-order correction to wave functions of harmonic oscillator are deduced.     

Mode Competition and Self-Consistent Simulation of a Second Harmonic Gyrotron Oscillator

A nonlinear numerical simulation of a second harmonic gyrotron oscillator is presented in this paper. Mode competition from a competing fundamental mode is investigated. In addition, a self-consistent nonlinear theoretical model, including the effect of velocity spread, is applied to analyze the second harmonic gyrotron. A series of numerical calculations is carried out for different electron beam parameters. The results are compared with that of calculations using the cold-cavity, fixed field approximation and with experiments. Good agreement is found between our calculations and an experiment at Fukui University. The azimuthal polarization of the field and the choice of the electron beam radius are also discussed.     

一维无限深势阱定态波函数动量展开讨论

一维无限深势阱定态的动量概率分布存在两种"互相矛盾"的结论.为帮助学生理解两种结论的异同,锻炼严谨的科学作风和解决复杂问题的能力,笔者在量子力学理论教学中指导学生利用Python等现代科学计算工具,将一维无限深势阱定态波函数动量展开的讨论进行数值求解与可视化,绘制了不同观点以及不同情形下粒子的动量概率分布图.以此为基础讨论了不同计算结果的物理含义,揭示了两种结论的内在联系.此创新性的举措不仅有效加深了学生对于该重要问题的理解,还促进了学生专业知识、科学能力和综合素质的全面培养.     

二维无限深方势阱中粒子运动的路径积分解法

用路径积分的方法计算了二维无限深方势阱中粒子的传播子,并由传播函数推导出二维无限深方势阱中粒子的波函数和能量,进一步体现了路径积分与其他经典量子化方法的等价性,反映了路径积分应用于难以处理的量子力学问题的价值.