Behavioral Differences of a Time-Dependent Harmonic Oscillator in Commutative Space and Non-Commutative Phase Space

In this article, behavioral differences of time-dependent harmonic oscillator in Commutative space and Non-Commutative phase space have been investigated. The considered harmonic oscillator has a time-dependent angular frequency and mass which are function of time. First, the time-dependent harmonic oscillator is studied in commutative space, then similar calculation is done for considered harmonic oscillator in Non-Commutative phase space. During this article method of Lewis–Riesenfeld dynamical invariant has been employed.     

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.     

Quantum harmonic oscillator with time-dependent mass and frequency

The quantum harmonic oscillator with time-dependent mass and frequency is analyzed by using the canonical transformation method. The varying mass and frequency of the system are reduced to constant mass and frequency, and the corresponding eigenvalues and eigenvectors are derived. The exact time-dependent coherent state of the harmonic oscillator is constructed and shown to be equivalent to the squeezed state. Damped harmonic oscillators with different frictions and forced time-dependent harmonic oscillators are also discussed.     

Schwinger action principle via linear quantum canonical transformations

We have applied the Schwinger action principle to general one-dimensional (1D), time-dependent quadratic systems via linear quantum canonical transformations, which allowed us to simplify the problems to be solved by this method. We show that while using a suitable linear canonical transformation, we can considerably simplify the evaluation of the propagator of the studied system to that for a free particle. The efficiency and exactness of this method is verified in the case of the simple harmonic oscillator. This technique enables us to evaluate easily and immediately the propagator in some particular cases such as the damped harmonic oscillator, the harmonic oscillator with a time-dependent frequency, and the harmonic oscillator with time-dependent mass and frequency, and in this way the propagator of the forced damped harmonic oscillator is easily calculated without any approach. PACS 02.30.Xx, 03.65.-w, 03.65.Ca     

二维各向同性变频率谐振子的超对称性及其不变量的超对称量子力学精确解

采用超对称量子力学与不变量相结合的方法讨论了二维各向同性变频率谐振子,给出了二维各向同性变频率谐振子的不变量,采用超对称量子力学方法精确求解了不变量的本征值和本征函数,并且给出了当频率恒定时,二维常频率谐振子的本征值和本征函数的精确解.最后对不变量的超对称性进行了讨论.     

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.     

Exact solution of the time-dependent harmonic plus an inverse harmonic potential with a time-dependent electromagnetic field

In this paper, the problem of the charged harmonic plus an inverse harmonic oscillator with time-dependent mass and frequency in a time-dependent electromagnetic field is investigated. It is reduced to the problem of the inverse harmonic oscillator with time-independent parameters and the exact wave function is obtained.     

Stochastic Quantization of Time-Dependent Systems by the Haba and Kleinert Method

The stochastic quantization method recently developed by Haba and Kleinert is extended to non-autonomous mechanical systems, in the case of the time-dependent harmonic oscillator. In comparison with the autonomous case, the quantization procedure involves the solution of a nonlinear, auxiliary equation. Using a rescaling transformation, the Schrö-din-ger equation for the time-dependent harmonic oscillator is obtained after averaging of a classical stochastic differential equation.     

Stochastic Quantization of Time-Dependent Systems by the Haba and Kleinert Method

The stochastic quantization method recently developed by Haba and Kleinert is extended to non-autonomous mechanical systems, in the case of the time-dependent harmonic oscillator. In comparison with the autonomous case, the quantization procedure involves the solution of a nonlinear, auxiliary equation. Using a rescaling transformation, the Schrödinger equation for the time-dependent harmonic oscillator is obtained after averaging of a classical stochastic differential equation.     

Exact Time-Dependent Wave Functions of a Confined Time-Dependent Harmonic Oscillator with Two Moving Boundaries

By applying the standard analytical techniques of solving partial differential equations, we have obtained the exact solution in terms of the Fourier sine series to the time-dependent Schrödinger equation describing a quantum one-dimensional harmonic oscillator of time-dependent frequency confined in an infinite square well with the two walls moving along some parametric trajectories. Based upon the orthonormal basis of quasi-stationary wave functions, the exact propagator of the system has also been analytically derived. Special cases like (i) a confined free particle, (ii) a confined time-independent harmonic oscillator, and (iii) an aging oscillator are examined, and the corresponding time-dependent wave functions are explicitly determined. Besides, the approach has been extended to solve the case of a confined generalized time-dependent harmonic oscillator for someparametric moving boundaries as well.     

具有含时平方反比项的谐振子的路径积分求解

使用含时坐标变换把一个质量含时的带有平方反比项的谐振子系统变换为对应的质量不含时 谐振子系统.利用变换前、后传播子之间的关系并通过Feynman路径积分的方法求出质量含时 的带有平方反比项的谐振子系统的传播子与精确波函数.并对包含有更多的附加势的谐振子 系统进行了讨论. 关键词： 谐振子 坐标变换 路径积分 传播子     

Gauge covariant construction of the coherent states for a time-dependent harmonic oscillator by algebraic dynamical method

Time-dependent coherent states for a time-dependent harmonic oscillator are constructed in the framework of algebraic dynamics. These coherent states are gauge-covariant, and its time evolution is governed only by the solutions of a linear differential equation which describes the motion of the corresponding classical timedependent harmonic oscillator. Its non-classical and quantum statistical properties can thus be controlled by a proper choice of the frequency of the harmonic oscillator. Our coherent states reduce to Glauber coherent states in the case as the frequency is independent of time.     

Time-Dependent 2D Harmonic Oscillator in Presence of the Aharanov-Bohm Effect

We use the Lewis-Riesenfeld theory to determine the exact form of the wavefunctions of a two-dimensionnal harmonic oscillator with time-dependent mass and frequency in presence of the Aharonov-Bohm effect (AB). We find that the auxiliary equation is independent of the AB magnetic flux. In the particular case of quantized AB magnetic flux the wavefunctions coincide exactly with the wavefunctions of the 2D time-dependent harmonic oscillator. PACS: 03.65Ge; 03.65Fd; 03.65Bz     

Propagator for a Time-Dependent Damped Harmonic Oscillator with a Force Quadratic in Velocity

The propagator for a time-dependent damped harmonic oscillator with a force quadratic in velocity isobtained by making a specific coordinate transformation and by using the method of time-dependent invariant.     

More exact invariants for the time-dependent harmonic oscillator

We derive an infinite number of new invariants and related auxiliary equations for the time-dependent harmonic oscillator.     

Forced Time-Dependent Harmonic Oscillators in Non-Commutative Space

For the time-dependent harmonic oscillator and generalized harmonic oscillator with or without external forces in non-commutative space, wave functions, and geometric phases are derived using the Lewis-Riesenfeld invariant. Coherent states are obtained as the ground state of the forced system. Quantum fluctuations are calculated too. It is seen that geometric phases and quantum fluctuations are greatly affected by the non-commutativity of the space.     

Ambiguities Appearing in the Study of Time-Dependent Constants of Motion for the One-Dimensional Harmonic Oscillator

A family of time-dependent constants of motionfor the one-dimensional harmonic oscillator is derived.The relation between constants of motion, Lagrangian,and Hamiltonian is described. A well-defined time-dependent Lagrangian (for whichEuler–Lagrange equations and Legendretransformation are fully satisfied) is not uniquelydetermined.     

Generalized path-integral solution and coherent states of the harmonic oscillator in D-dimensions

We construct a general form of propagator in arbitrary dimensions and give an exact wavefunction of a time- dependent forced harmonic oscillator in D（D ≥ 1） dimensions. The coherent states, defined as the eigenstates of annihilation operator, of the D-dimensional harmonic oscillator are derived. These coherent states correspond to the minimum uncertainty states and the relation between them is investigated.     

Dynamical invariants and time-dependent harmonic systems

A simplified algebraic derivation of the dynamical (Lewis) invariant for the time-dependent harmonic oscillator is presented. The treatment is based on the concept of the dynamical algebra.     

BERRY PHASES IN THE QUANTUM STATE OF THE ISOTROPIC HARMONIC OSCILLATOR WITH TIME-DEPENDENT FREQUENCY AND BOUNDARY CONDITIONS

The problem of the isotopic harmonic oscillator of time-dependent frequency confined in a spherical box with time-dependent radius is studied. We show that the exact solution and the Lewis invariant operator can be obtained by performing two consecutive gauge transformations on the time-dependent Schr?dinger equation. On the basis of the exact solution the non-adiabatic Berry phases for the system are calculated.