Quantum probability applied to the damped harmonic oscillator
Hans Maassen
在这门入门课程中，我们将勾勒量子机率的架构，以讨论开放量子系统，特别是阻尼谐振子。
In this introductory course we sketch the framework of quantum probability in order to discuss open quantum systems, in particular the damped harmonic oscillator.
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Coherent and trajectory-coherent states of a damped harmonic oscillator
本文构造了阻尼谐振子的相干态和非相干相干态。我们研究了这些状态的性质。
Decoherence of Quantum Damped Oscillators
研究了两个模型振子中的量子耗散，Caldirola-Kanai（CK）振子是一个具有一个自由度的开放系统，Bateman-Feshbach-Tikochinsky（BFT）振子是一个具有两个自由度的封闭系统.虽然这些振子描述了相同的经典阻尼运动，但CK振子保持了量子相干性，而BFT振子的阻尼子系统同时表现出量子退相干和经典关联。此外，BFT振荡器的放大子系统表现出相同程度的量子退相干和经典关联。
Quantum dynamics of a damped deformed oscillator
研究了量子形变振子与环境的相互作用，导出了主方程，其形式强烈依赖于形变的类型。
The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a dam…
We solve the optimal quantum limit of probing a classical force exactly by a damped oscillator initially prepared in the factorized squeezed state. The memory effects of the thermal bath on the oscillator evolution are investigated. We show that the optimal force sensitivity obtained by the quantum estimation theory approaches to zero for the non-Markovian bath, whereas approaches to a finite non-zero value for the Markovian bath as the energy of the damped oscillator goes to infinity.
We analyse the properties of a strongly-damped quantum harmonic oscillator by means of an exact diagonalisation of the full Hamiltonian, including both the oscillator and the reservoir degrees of freedom to which it is coupled. Many of the properties of the oscillator, including its steady-state properties and entanglement with the reservoir can be understood and quantified in terms of a simple probability density, which we may associate with the ground-state frequency spectrum of the oscillator.
Radiation reaction and quantum damped harmonic oscillator
以Klein-Gordon场作为谐振子的环境，采用一种处理量子耗散系统的新方法（最小耦合法），研究了量子阻尼谐振子的量子动力学和辐射反应.应用微扰方法，得到了表明能量在振子、库和量子真空之间流动的跃迁几率
Quantum damped oscillator I: dissipation and resonances
阻尼谐振子的量子化导致所谓的贝特曼对偶系统。相应的贝特曼哈密顿量，是一个自伴算子，显示离散家庭的复杂的特征值。我们证明了它们对应于能量本征向量的极点和相应的复能量平面上的预解算子。因此，相应的广义本征向量可以解释为共振态，这是负责阻尼谐振子的不可逆量子动力学。
Memory in a nonlocally damped oscillator
分析了阻尼振子的新运动方程。它不同于标准的一个阻尼项，这是非本地的时间，因此，它产生了一个系统的记忆。进行了经典和量子分析。这种非定域系统的特点是它打破了经典哈密顿动力学和相应的量子传播子的局部组成。
The phase-space path-integral approach to the damped harmonic oscillator is analyzed beyond the Markovian approximation. It is found that pairs of nonclassical trajectories contribute to the path-integral representation of the Wigner propagating function. Due to the linearity of the problem, the sum coordinate of a pair still satisfies the classical equation of motion. Furthermore, it is shown that the broadening of the Wigner propagating function of the damped oscillator arises due to the time-nonlocal interaction mediated by the heat bath.
The quantum dynamics of a damped harmonic oscillator is investigated in the presence of an anisotropic heat bath. The medium is modeled by a continuum of three dimensional harmonic oscillators and anisotropic coupling is treated by introducing tensor coupling functions. Starting from a classical Lagrangian, the total system is quantized in the framework of the canonical quantization. Following Fano technique, Hamiltonian of the system is diagonalized in terms of creation and annihilation operators that are linear combinations of the basic dynamical variables. Using the diagonalized Hamiltonian…
Density operator and entropy of the damped quantum harmonic oscillator
在开放量子系统的Lindblad理论框架下，从主方程出发，导出了阻尼谐振子的密度算符表达式。进而得到了系统的冯诺伊曼熵和有效温度。一个状态的特征在于由一个维格纳分布函数，这是高斯的形式被发现仅依赖于分布函数的方差的熵。
Quantum damped oscillator II: Bateman's Hamiltonian vs. 2D Parabolic Potential Barrier
我们表明，量子贝特曼的系统中产生的阻尼谐振子的量子化是等价的量子问题与二维抛物势垒也被称为二维倒各向同性振荡器。结果表明，该系统显示的家庭复特征值对应的极点的解析延续的预解算子的复能量平面。结果表明，这种表示比Blasone和Jizba最近使用的双曲表示更合适。
The quantum dynamics of a damped and forced harmonic oscillator is investigated in terms of a Lindblad master equation. Elementary algebraic techniques are employed allowing for example to analyze the long time behavior, i.e. the quantum limit cycle. The time evolution of various expectation values is obtained in closed form as well as the entropy and the Husimi phase space distribution. We also discuss the related description in terms of a non-Hermitian Hamiltonian.
In this letter we show that it is not possible to set up a canonical quantization for the damped harmonic oscillator using the Bateman lagrangian. In particular, we prove that no square integrable vacuum exists for the {\em natural} ladder operators of the system, and that the only vacua can be found as distributions. This implies that the procedure proposed by some authors is only formally correct, and requires a much deeper analysis to be made rigorous.
Wave-particle duality in the damped harmonic oscillator
阻尼谐振子的量子化被认为是为物理学家介绍量子概率论的主要内容。为此，我们以物理学（研究生）学生为入门级，解释量子概率论的物理直觉和动机，有时压倒性的数学机制。正文从海森堡和薛定谔的观点出发，从（无阻尼）谐振子的量子化开始。我们将展示如何处理的特殊情况下的量子概率量子化过程：第二个量子化函子。然后，我们应用第二量子化函子的阻尼谐振子和解释的量子动力学的位置和能量运营商作为随机过程。
A symmetric treatment of damped harmonic oscillator in extended phase space
量子统计力学的扩展相空间（EPS）公式把普通相空间坐标放在同一个基础上，从而允许确定与这些坐标共轭的正则动量。扩展拉格朗日量和扩展哈密顿量的定义在EPS中的相同的程序，为普通拉格朗日量和哈密顿量。普通相空间和它们的共轭动量的结合展示了粒子和它们的镜像的共同演化。对于阻尼谐振子，在EPS中得到的演化方程是这样的，即由实际谐振子耗散的能量以相同的速率被镜像谐振子吸收，使得整个系统作为保守系统。本文利用EPS方法，在扩展相空间中，通过一个简单的扩展正则变换，得到了由Bateman首先提出的阻尼谐振子的对偶哈密顿量。推广的正则变换能够将实际振子和镜像振子的阻尼系统转化为无阻尼系统，并将演化方程转化为简单形式。求解所得方程，得到阻尼振子及其镜像的本征值和本征函数。结果与Bateman的结果一致。最后，对上述系统的不确定度关系进行了检验.
We obtain the tomogram of squeezed correlated states of a quantum parametric damped oscillator in an explicit form. We study the damping within the framework of the Caldirola--Kanai model and chose the parametric excitation in the form of a very short pulse simulated by a delta-kick of frequency; the squeezing phenomenon is reviewed. The cases of strong and weak damping are investigated.
We investigate the exact dynamics of the damped quantum harmonic oscillator under the (un)correlated initial conditions. The master equation is generalized to the cases of the arbitrary factorized state and/or Gaussian state. We show that the variances of the factorized Gaussian state do not sensitively depend on the initial oscillator-bath correlation, which however can remarkably affect the mean values even at high temperature. We also illustrate that the correlations among the factorized states still give rise to the initial dips during the purity evolutions, which can be smoothed out by in…
Quantizing the damped harmonic oscillator
考虑经典阻尼谐振子的费米量子化。在过去的研究中，作者将dho的相空间加倍，以使系统在每个时刻都是封闭的。对于无限维的相空间，这种方法需要为每个时间构造CAR代数的表示。我们表明，酉膨胀的收缩半群控制系统的动力学是一个逻辑扩展的加倍过程，它允许一个，以避免与以前的方法遇到的数学困难。
Misbelief and misunderstandings on the non--Markovian dynamics of a damped harmonic oscillator
本文用阻尼谐振子的精确解来讨论它的开放动力学中常被误解的一些有关方面。我们比较两种不同的近似都被称为旋转波近似。通过一个具体的例子，我们澄清了一些有关的非马尔可夫动力学，非Lindblad型动力学，和积极的密度矩阵。
The investigation of quantum-classical correspondence may lead to gain a deeper understanding of the classical limit of quantum theory. We develop a quantum formalism on the basis of a linear-invariant theorem, which gives an exact quantum-classical correspondence for damped oscillatory systems that are perturbed by an arbitrary force. Within our formalism, the quantum trajectory and expectation values of quantum observables are precisely coincide with their classical counterparts in the case where we remove the global quantum constant h from their quantum results. In particular, we illustrate…
Disagreement between correlations of quantum mechanics and stochastic electrodynamics in the damped parametric oscillator
利用量子力学和随机电动力学（SED）半经典理论计算了阻尼非简并参量振荡器的腔内和外三阶关联。这两种理论得到的结果有很大的不同，量子力学的关联在系统的非线性耦合常数下是立方的，而SED的关联在相同的常数下是线性的。特别是，这两种理论之间的差异至少存在于介观制度。当包括实际阻尼时，它们也存在。这种差异说明了量子力学和连续变量的隐变量理论之间的区别。
Quantum decoherence of the damped harmonic oscillator
在开放量子系统的Lindblad理论框架下，我们确定了谐振子与热浴相互作用的量子退相干度。结果表明，该系统表现出量子退相干，并且随着时间的推移，退相干越来越显著。我们还计算了退相干时间，并表明它具有相同的规模后，热涨落变得可比量子涨落的时间。
Nowadays, two of the most prospering fields of physics are quantum computing and spintronics. In both, the loss of information and dissipation plays a crucial role. In the present work we formulate the quantization of the dissipative oscillator, which aids understanding of the above mentioned, and creates a theoretical frame to overcome these issues in the future. Based on the Lagrangian framework of the damped spring system, the canonically conjugated pairs and the Hamiltonian of the system are obtained, by which the quantization procedure can be started and consistently applied. As a result,…
The Quantum state diffusion model and the driven damped nonlinear oscillator
考虑一个受驱动的阻尼非谐振子，它经典地导致一个非稳态和滞后。该系统的量子对应物在稳态下具有精确的解析解，该解析解不显示任何双稳态或滞后。我们用量子态扩散理论来描述这个系统，并提供了一个新的视角，在量子区的滞后缺乏，从而详细研究量子到经典的转变。该分析也是相关的测量一个周期性驱动的电子在潘宁陷阱，其中已观察到滞后。
Exact Diagonalization of Two Quantum Models for the Damped Harmonic Oscillator
阻尼谐振子是量子力学中研究耗散的重要工具。然而，尽管它的简单性，这个系统已经引起了一些近似的有效性和关系，更精细的描述值得一个彻底的调查。在这项工作中，我们采用的方法，使我们能够对角化正是经常采用的耗散哈密顿在文献中。使用这种方法，我们推导出的有效性条件的行波近似（RWA），并显示这种近似的描述涉及到更一般的。我们还证明了耗散相干态的存在与RWA密切相关。最后，通过阻尼振荡器的动力学评价，我们注意到耗散模型的一个重要性质，在以前的作品中没有得到适当的考虑，即，新的约束的必要性，可因子化的初始条件的应用。
Revisiting the damped quantum harmonic oscillator
本文重新分析了量子阻尼谐振子，通过引入储层连续统模型、Fano技术的精确对角化以及热场技术，恢复了已知与新结果，并从头证明了振荡器弛豫至平均力Gibbs态。
We derive an exact non-Markovian master equation that generalizes the previous work [Hu, Paz and Zhang, Phys. Rev. D {\bf 45}, 2843 (1992)] to damped harmonic oscillators with time-varying parameters. This is achieved by exploiting the linearity of the system and operator solution in Heisenberg picture. Our equation governs the non-Markovian quantum dynamics when the system is modulated by external devices. As an application, we apply our equation to parity kick decoupling problems. The time-dependent dissipative coefficients in the master equation are shown to be modified drastically when the…
Damped quantum harmonic oscillator
在开放量子系统的Lindblad理论框架下，研究了谐振子的阻尼。本文对阻尼量子振子主方程中的量子力学扩散系数的基本约束进行了推广，明确给出了林德布拉德方程的薛定谔和海森堡表示.在此基础上，我们证明了文献中所用的阻尼量子振子的各种主方程都是林德布拉德方程的特例，并且这些方程中的大多数都不满足量子力学扩散系数的约束条件.利用Lindblad主方程的特征函数，得到了坐标和动量的前两阶矩的解析表达式。主方程转化为Fokker-Planck方程的准概率分布。对Glauber $P$表示、反正规序$Q$表示和Wigner $W$表示进行了比较研究.证明了用这些表象求出的阻尼谐振子的方差是相同的。通过求解稳态下的Fokker-Planck方程，证明了准概率分布是二维高斯分布，其宽度由扩散系数决定。密度矩阵通过生成函数表示，生成函数通过求解从主方程导出的时间相关的线性偏微分方程获得。提供了密度矩阵的特定初始条件的说明性示例。
The damped harmonic oscillator in deformation quantization
在形变量子化的框架下，提出了阻尼谐振子量子化的一种新方法。量子化是在薛定谔图像通过一个修改的“泊松括号”诱导的星积。我们确定了阻尼区的本征态，并计算了无阻尼谐振子耗散后的态间跃迁几率。
We revisit quantum dynamics of the damped and driven nonlinear oscillator. In the classical case this system has two stationary solutions (the limit cycles) in the certain parameter region, which is the origin of the celebrated bistability phenomenon. The quantum-classical correspondence for the oscillator dynamics is discussed in details.
We consider position-dependent mass (PDM) Lagrangians/Hamiltonians in their standard textbook form, where the long-standing \emph{gain-loss balance} between the kinetic and potential energies is kept intact to allow conservation of total energy (i.e., $L=T-V$, $H=T+V$, and $dH/dt=dE/dt=0$). Under such standard settings, we discuss and report on $n$-dimensional PDM damped harmonic oscillators (DHO). We use some $n$-dimensional point canonical transformation to facilitate the linearizability of their $n$-PDM dynamical equations into some $n$-linear DHOs' dynamical equations for constant mass set…
Kraus representation of damped harmonic oscillator and its application
根据定义，受环境影响的谐振子的Kraus表示，被建模为振幅阻尼或相位阻尼，直接由简单的算子代数解给出。作为例子和应用，我们首先给出了一个单量子比特的Kraus表示，其计算基态定义为玻色子真空态和单粒子数态。我们进一步讨论了以玻色子奇、偶相干态为计算基态的量子比特的环境效应。比较了环境对两种不同计算基定义的纠缠量子比特的影响。
A variety of dynamics in nature and society can be approximately treated as a driven and damped parametric oscillator. An intensive investigation of this time-dependent model from an algebraic point of view provides a consistent method to resolve the classical dynamics and the quantum evolution in order to understand the time-dependent phenomena that occur not only in the macroscopic classical scale for the synchronized behaviors but also in the microscopic quantum scale for a coherent state evolution. By using a Floquet U-transformation on a general time-dependent quadratic Hamiltonian, we ex…
Lindblad master equation for the damped harmonic oscillator with deformed dissipation
在开放量子系统的Lindblad理论框架下，导出了基于变形代数的量子谐振子与耗散环境，特别是与热浴相互作用的主方程.观测量的运动方程强烈依赖于形变函数。在热浴温度为零的情况下，在小变形参数的极限下，计算了数算符和平方数算符的期望值。得到了密度矩阵方程的稳态解，并证明了其与变形无关。
Uncertainty, entropy and decoherence of the damped harmonic oscillator in the Lindblad theory of open quantum systems
在开放量子系统的Lindblad理论框架下，得到了阻尼谐振子的密度算符、冯诺伊曼熵和有效温度的表达式.一个状态的特征在于由一个维格纳分布函数，这是高斯的形式被发现仅依赖于分布函数的方差的熵。我们给出了一系列不等式，将不确定性与冯诺伊曼熵和线性熵联系起来。我们分析了状态纯度的条件，并表明，对于一个特殊的选择的扩散系数，相关相干态（压缩相干态）是唯一的状态，保持纯的所有时间在所考虑的系统的演化。这些态也是在环境存在下演化过程中最稳定的态，在描述环境诱导退相干中起着重要的作用。
Damped harmonic oscillator: pure states of the bath and exact master equations
本文比较了热浴中耦合谐振子在巨正则系综、数态和相干态三种初始态下的时间演化。结果表明，对于广泛的一类数态，振子的行为与平衡浴的情况相似。如果浴模最初处于相干态，则振子坐标和动量的方差以及与浴的纠缠度渐进地接近与零温度下振子相同的值，并且平均坐标和动量表现出布朗态-类行为。我们推导出一个精确的主方程的振子的特征函数有效的任意因式分解的初始条件。在平衡浴的情况下，这个方程减少到一个方程的Hu-Paz-Zhang型，而对于相干态浴，它导致一个确切的随机主方程与乘性噪声。
Effective Description of the Quantum Damped Harmonic Oscillator: Revisiting the Bateman Dual System
本文提出基于矩量子力学的阻尼谐振子量子化方案，利用巴特曼对偶模型，所得描述与林德布拉德主方程高度一致，为开放量子系统提供有效且普适的工具。
We model a quantum system coupled to an environment of damped harmonic oscillators by following the approach of Caldeira-Leggett and adopting the Caldirola-Kanai Lagrangian for the bath oscillators. In deriving the master equation of the quantum system of interest (a particle in a general potential), we show that the potential is modified non-trivially by a new inverted harmonic oscillator term, induced by the damping of the bath oscillators. We analyze numerically the case of a particle in a double-well potential, and find that this modification changes both the rate of decoherence at short t…
The physics of quantum electromagnetism in an absorbing medium is that of a field of damped harmonic oscillators. Yet until recently the damped harmonic oscillator was not treated with the same kind of formalism used to describe quantum electrodynamics in a arbitrary medium. Here we use the techniques of macroscopic QED, based on the Huttner--Barnett reservoir, to describe the quantum mechanics of a damped oscillator. We calculate the thermal and zero-point energy of the oscillator for a range of damping values from zero to infinity. While both the thermal and zero-point energies decrease with…
Closed-Form Expectation Values of the Damped Kerr Oscillator
该研究针对相干态下的阻尼克尔振荡器，从林德布拉德主方程精确解出发，推导出期望值的闭合形式表达式，无需数值积分和福克空间截断，经模拟验证，此解为数值模拟提供有效替代，还指出李代数结构扩展或助于求解更广泛的开放量子光学系统。
General Solution of the Quantum Damped Harmonic Oscillator
本文给出了量子阻尼谐振子的一般解。
The dynamics of the next quantum jump for a qubit [two level system] coupled to a readout resonator [damped driven harmonic oscillator] is calculated. A quantum mechanical treatment of readout resonator reveals non exponential short time behavior which could facilitate detection of the state of the qubit faster than the resonator lifetime.
Purity and decoherence in the theory of a damped harmonic oscillator
对于Karrlein和Grabert导出的阻尼谐振子微观模型的广义主方程，给出了不同初始条件和不同阻尼类型（包括欧姆、Drude和弱耦合情况，Agarwal和Weidlich-Haake模型）下的态纯度条件.结果表明，保持纯态的态是时间方差为常数的压缩态。对于纯态，也得到了与这些主方程对应的广义非线性Schr”odinger型方程.然后在开放量子系统的Lindblad理论框架下讨论了阻尼谐振子的态的纯度条件。对于一个特殊的环境系数的选择，具有恒定的方差和协方差的相关相干态被证明是唯一的状态，保持纯的所有时间在所考虑的系统的发展。在Karrlein-Grabert和Lindblad模型中，以及在所考虑的特殊模型中，给出了熵产生率的表达式，并证明了在时间上保持其纯性的状态也是熵产生最小的状态，因此，它们是在环境存在下演化的最稳定的状态，在描述退相干现象中起着重要的作用。
Damped harmonic oscillators in the holomorphic representation
量子动力学半群被应用于研究玻色子和费米子谐振子的时间演化。利用全纯表示导出了描述这些系统状态的密度矩阵的显式表达式。玻色子和费米子的自由度，然后放在一起，形成一个超对称振荡器，确保相应的动力学方程的超对称不变性的条件明确推导。
Damped driven coupled oscillators: entanglement, decoherence and the classical limit
只要阻尼常数足够大，即使在很低的温度和平均光子数下，（二能级）里德伯原子与耗散QED腔场的相互作用也可以用经典或量子力学的方法描述。研究了一个解析可解模型的量子-经典边界、纠缠和退相干，该模型类似于原子-腔系统，其中原子（场）由一个（受驱动和阻尼的）谐振子表示.纠缠的最大值取决于初始状态和耗散率耦合常数比。虽然在原始模型中原子熵从未明显增长（对于大耗散率），但在我们的模型中，它在下降之前达到最大值。虽然这两种模型预测的纠缠和耗散的小值，为固定的时间的顺序的耦合常数和大的耗散率的倒数，这些数量减少得更快，作为一个函数的耗散率的耦合常数，在我们的模型。
The quantum dynamics of a damped and forced harmonic oscillator described by a Lindblad master equation is analyzed. The master equation is converted into a matrix-vector representation and the resulting non-Hermitian Schrödinger equation is solved by Lie-algebraic techniques allowing the construction of the general solution for the density operator.
Fractional Driven Damped Oscillator
本文研究分数阶阻尼振子在振荡外力驱动下的共振行为，发现可通过调节阻尼系数或分数阶导数阶数来操控共振。
We obtain analytic solutions to various models of dissipation of the quantum harmonic oscillator, employing a simple method in the Wigner function Fourier transform description of the system; and study as an exemplification, the driven open quantum harmonic oscillator. The environmental models we use are based on optical master equations for the zero and finite temperature bath and whose open dynamics are described by a Lindblad master equation, and also we use the Caldeira-Leggett model for the high temperature limit, in the the under damped an the over damped case. Under the Wigner Fourier t…
In this paper, it is proposed a quantization procedure for the one-dimensional harmonic oscillator with time-dependent frequency, time-dependent driven force, and time-dependent dissipative term. The method is based on the construction of dynamical invariants previously proposed by the authors, in which fundamental importance is given to the linear invariants of the oscillator.
A Canonical Approach to the Quantization of the Damped Harmonic Oscillator
基于自由度加倍方法，提出了一种研究量子力学阻尼谐振子的新正则方法。显式表达式拉格朗日的基本模式的问题，表征向前和向后的时间传播。一个哈密顿分析，显示等效的拉格朗日方法，也做了。在此基础上，讨论了模型的量子化。
We numerically study the quantum dynamics of a $\mathcal{PT}$-symmetric kicked harmonic oscillator. We observe that directed current of momentum and ballistic diffusion of energy coexist under the non-resonant conditions, whereas both the momentum and energy oscillate as damped cosine functions with identical frequencies under the resonant conditions. The research shows that the directed current of momentum and ballistic diffusion of energy arise from nearest-neighbor hopping between momentum eigenstates with the non-Hermitian driving, while the damped oscillations of momentum and energy origi…
Driven harmonic oscillator as a quantum simulator for open systems
我们从理论上证明了如何用受迫谐振子作为非马尔可夫阻尼谐振子的量子模拟器。在一般的框架下，结果表明，可以使用一个封闭的系统作为开放量子系统的模拟器。量子模拟器基于具有适当定制的电场脉冲的闭合谐振子的受控驱动器的集合。利用开放系统的谱密度信息对封闭振子的驱动进行平均，得到了阻尼谐振子的非马尔可夫动力学。我们认为单个被困离子作为一个特定的物理实现的模拟器，我们展示了如何模拟器的方法揭示了新的物理洞察到开放的系统动力学，例如特征量子力学非马尔可夫振荡行为的阻尼振荡器的能量，通常由非Lindblad型主方程，可以有一个简单的半经典解释。
Moment expansion method for composite open quantum systems including a damped oscillator mode
本文提出一种矩展开数值方法，通过求解正交算符矩的演化方程来计算含阻尼振子模式的复合开放量子系统中目标系统的约化密度矩阵，在光力系统中显著降低计算成本，并发现位置基可缓解数值敏感性问题。
Caldirola-Kanai Oscillator in Classical Formulation of Quantum Mechanics
在基于层析成像方案的量子力学公式框架内，引入了量子阻尼振子的正交分布。阻尼振子相干态和Fock态的几率分布分别用高斯多项式和厄米多项式表示。
Valtancoli in his paper entitled [P. Valtancoli, Canonical transformations, and minimal length J. Math. Phys. 56, 122107 (2015)] has shown how the deformation of the canonical transformations can be made compatible with the deformed Poisson brackets. Based on this work and through an appropriate canonical transformation, we solve the problem of one dimensional (1D) damped harmonic oscillator at the classical limit of the Snyder-de Sitter (SdS) space. We show that the equations of the motion can be described by trigonometric functions with frequency and period depending on the deformed and the …
A recent paper [J. Math. Phys. {\bf 59}, 082105 (2018)] constructs a Hamiltonian for the (dissipative) damped harmonic oscillator. We point out that non-Hermiticity of this Hamiltonian has been ignored to find real discrete eigenvalues which are actually non-real. We emphasize that non-Hermiticity in Hamiltonian is crucial and it is a quantal signature of dissipation.
We return to the description of the damped harmonic oscillator by means of a closed quantum theory with a general assessment of previous works, in particular the Bateman-Caldirola-Kanai model and a new model recently proposed by one of the authors. We show the local equivalence between the two models and argue that latter has better high energy behavior and is naturally connected to existing open-quantum-systems approaches.
In this chapter we treat the quantum damped harmonic oscillator, and study mathematical structure of the model, and construct general solution with any initial condition, and give a quantum counterpart in the case of taking coherent state as an initial condition. This is a simple and good model of Quantum Mechanics with dissipation which is important to understand real world, and readers will get a powerful weapon for Quantum Physics.
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