Three useful bounds in quantum mechanics - easily obtained by Wiener integration
Hajo Leschke, Rainer Ruder
在一个合理的自足和明确的介绍，我们说明了效率的费曼-卡茨公式的严格推导的三个不等式的兴趣在非相对论性量子力学。
In a reasonably self-contained and explicit presentation we illustrate the efficiency of the Feynman-Kac formula for the rigorous derivation of three inequalities of interest in non-relativistic quantum mechanics.
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We study a (relativistic) Wiener process on a complexified (pseudo-)Riemannian manifold. Using Nelson's stochastic quantization procedure, we derive three equivalent descriptions for this problem. If the process has a purely real quadratic variation, we obtain the one-sided Wiener process that is encountered in the theory of Brownian motion. In this case, the result coincides with the Feyman-Kac formula. On the other hand, for a purely imaginary quadratic variation, we obtain the two-sided Wiener process that is encountered in stochastic mechanics, which provides a stochastic description of a …
Emergence of a Wiener process as a result of the quantum mechanical interaction with a macroscopic medium
我们分析了一个修改后的版本的科尔曼-赫普模型，这是能够考虑到传入的粒子和线性阵列组成的$N$自旋1/2系统之间的能量交换过程。我们带来光的维纳耗散过程中存在的弱耦合，宏观（$N \rightarrow \infty$）限制。在这样一个极限中，在整个希尔伯特空间的一个受限部分中，粒子经历一种布朗运动，而自旋阵列的自由哈密顿量则充当一个维纳过程。对自旋系统的哈密顿量的谱不作任何假设，也不计算其状态的部分迹。的随机过程的外观的机制进行了讨论，并在文献中的其他值得注意的例子进行对比。强调了与van Hove的“$λ^2 T$极限”的联系。
Reply to ``Comment on `On the inconsistency of the Bohm-Gadella theory with quantum mechanics'''
在这个答复中，我们表明，当我们应用标准分布理论的李普曼-施温格方程，得到的空间的测试功能将符合哈迪公理，只有经典的结果佩利和维纳，Gelfand和Shilov，和理论的超分布是错误的。同时，我们指出了在量子力学中构造被操纵的希尔伯特空间的"标准方法“与在时间非对称量子理论中使用的方法之间的几个不同之处。
Quantum Mechanics from Stochastic Processes
本研究建立非相对论/相对论随机过程分别与薛定谔、克莱因-戈登方程解的一一对应，提出洛伦兹路径积分可定义为伊藤积分，为量子理论提供随机诠释。
Stochastic Mechanics and the Unification of Quantum Mechanics with Brownian Motion
本文用复平面旋转的 Wiener 过程统一布朗运动与非相对论量子力学，并借助二阶几何扩展到流形上的相对论随机理论，指出路径积分量子化要求由二次变差与仿射联络耦合导致的 Ito 对称性形变。
We show that quantum probabilities can be derived from statistical mechanics of classical fields. We consider Brownian motion in the space of fields and show that such a random field interacting with threshold type detectors produces clicks at random moments of time. And the corresponding probability distribution can be approximately described by the formalism of quantum mechanics. Hence, probabilities in quantum mechanics and classical statistical mechanics differ not so much as it is typically claimed. The temporal structure of the "prequantum random field" (which is the $L_2$-valued Wiener …
Quantum Mechanical Coherence, Resonance, and Mind
诺伯特·维纳和J.B.S.在20世纪30年代早期，哈尔科夫提出，量子理论对我们物质概念的深刻改变，为我们的思想以及其他经验的或类似心灵的性质开辟了道路，使它们在自然界中发挥作用，这种作用是因果互动的和有效的，而不是像经典力学所要求的那样纯粹是副现象的。这里描述了这一建议的数学基础，然后表明，通过赋予心灵在自然过程中的这种有效作用，我们对量子宇宙的感知的经典特征可以被看作是物种生存的进化压力的结果。
Quasi-hermitian Quantum Mechanics in Phase Space
我们使用标准的形变量子化方法：Groenewold星积和Wigner变换研究相空间中的准厄米量子力学。我们专注于假想刘维理论作为一个代表性的例子，准确的结果很容易得到。我们强调空间周期解，计算各种分布函数和相空间度量，并探讨它们之间的关系。
This paper proposes a very simple perturbative technique to calculate the low-lying eigenvalues and eigenstates of a parity-symmetric quantum-mechanical potential. The technique is to solve the time-independent Schroedinger eigenvalue problem as a perturbation series in which the perturbation parameter is the energy itself. Unlike nearly all perturbation series for physical problems, for the ground state this perturbation expansion is convergent and, even though the ground-state energy is in general not small compared with 1, the perturbative results are numerically accurate. The perturbation …
On the notion of phase in mechanics
相的概念在经典力学和量子力学中都扮演着重要的角色。但是什么是相呢？我们表明，如果我们定义相的概念相（！）空间中，人们可以非常容易和自然地恢复海森堡-外尔形式主义;这是使用庞加莱-嘉当不变量的性质来实现的，而不需要做任何量子假设。
In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefu…
Supersymmetry in Quantum Mechanics
本文对量子力学中的超对称性问题作了初步的介绍。我们用明确的例子证明，给定一个具有n个束缚态的量子力学中的可解问题，可以构造新的精确可解的n个哈密顿量，它们具有n-1，n-2，.，0绑定状态导出了超对称伙伴势的本征值、本征函数与散射矩阵之间的关系，并构造了一类无反射势。我们将求解一维谐振子问题的算子方法推广到一类形状不变的势。进一步，我们证明了，给定任何至少有一个束缚态的势，可以很容易地构造一个连续的具有相同特征值和s-矩阵的势参数族。本文还讨论了超对称激发的WKB近似（SWKB），结果表明，与通常的WKB不同，最低阶SWKB近似对形状不变势是精确的。最后，我们还利用超对称量子力学构造了新的精确可解周期势。
Quantum Uncertainty and Particle Trajectories: From Heisenberg and Schrödinger to Constructive Dynamical Models
本文区分量子不确定性关系与广义不确定性原理，通过费曼路径、纳尔逊随机力学、玻姆力学及卡茨动力学证明不确定性关系不排除底层轨迹描述，从而驳斥其否定微观轨迹的本体论解读。
This is the reply to the paper of Andrei Khrennikov arXiv:1505.04293 in which he expresses dissatisfaction with that the rough data in quantum experiments is not easily available and compares it with the open rough data in genetics. I try to explain why quantum experiments rough data is closed and why it differs radically from the biological. I also tried to answer the more thorny issue: is it possible to check quantum mechanics independently of other humans, e.g. trusting nobody.
A condition on the Hamiltonian of a time-dependent quantum mechanical system is derived which, if satisfied, implies optimal adiabaticity (defined below). The condition is expressed in terms of the Hamiltonian and in terms of the evolution operator related to it. Since the latter depends in a complicated way on the Hamiltonian, it is not yet clear how the condition can be used to extract useful information about the optimal Hamiltonian. The condition is tested on an exactly-soluble time-dependent problem (a spin in a magnetic field), where perfectly adiabatic evolution can be easily identified…
Formulations of magnetic monopoles in a Hilbert-space formulation of quantum mechanics require Dirac's quantization condition of magnetic charge, which implies a large value that can easily be ruled out for elementary particles by standard atomic spectroscopy. However, an algebraic formulation of non-associative quantum mechanics is mathematically consistent with fractional magnetic charges of small values. Here, spectral properties in non-associative quantum mechanics are derived, applied to the ground state of hydrogen with a magnetically charged nucleus. The resulting energy leads to new st…
TBTK is a software development kit for quantum mechanical calculations and is designed to enable the development of applications that investigate problems formulated on second-quantized form. It also enables method developers to create solvers for tight-binding, DFT, DMFT, quantum transport, etc., that can be easily integrated with each other. Both through the development of completely new solvers, as well as front and back ends to already well established packages. TBTK provides data structures tailored for second-quantization that will encourage reusability and enable scalability for quantum…
Time of arrival in quantum mechanics is discussed in two versions: the classical axiomatic "time of arrival operator" introduced by J. Kijowski and the EEQT method. It is suggested that for free particles the two methods may lead to the same result. On the other hand the EEQT method can be easily geometrized within the framework of Galilei-Newton general relativistic quantum mechanics developed by M. Modugno and collaborators, and it can be applied to non-free evolutions. The way of geometrization of irreversible quantum dynamics based on dissipative Liouville equation is suggested.
Non-Hamiltonian Commutators in Quantum Mechanics
首先揭示了量子算符的辛结构，然后利用它在量子力学中引入了广义非哈密顿括号。很容易认识到，量子经典系统是由这样一个括号的特定实现来描述的。根据以前的工作，这介绍了一个统一的方法来经典和量子经典非哈密顿动力学。为了说明使用非哈密顿算符，它显示了如何定义量子经典系统中的热力学约束。特别是，量子经典的Nosé-Hoover运动方程和相关的静态密度矩阵的推导。同时给出了Nosé-Hoover链和Nosé-Andersen（恒压恒温）动力学的非哈密顿算符。形式主义的观点进行了讨论。
The GMc-interpretation of Quantum Mechanics
GMC解释（引力-质量-光以最大速度c的运动）是一种与经典物理非常密切相关的量子力学的一致方法。在经典物理学、广义相对论、量子理论和热力学中，有几个假设是可以满足的。在这种解释中，粒子总是粒子，而不是波，许多悖论可以很容易地理解或避免。特别是，假设允许解释测量问题和无相互作用测量的概念;后者允许在不“触摸”它们的情况下找到对象，有时使用短语“在黑暗中看到”。此外，互补性，不确定性，退相干，局部性和现实主义的概念进行了研究。这种解释的一个关键性质是，在粒子进入实验之前，所有观测到的概率和干涉图样都是已知的。本体论的问题进行了讨论。
Symmetries are a key concept to connect mathematical elegance with physical insight. We consider measurement assemblages in quantum mechanics and show how their symmetry can be described by means of the so-called discrete bundles. It turns out that many measurement assemblages used in quantum information theory as well as for studying the foundations of quantum mechanics are entirely determined by symmetry; moreover, starting from a certain symmetry group, novel types of measurement sets can be constructed. The insight gained from symmetry allows us to easily determine whether the measurements…
Solutions of quaternionic quantum mechanics (QQM) are difficult to grasp, even in simple physical situations. In this article, we provide simple and understandable free particle quaternionic solutions, that can be easily compared to complex quantum mechanics (CQM). As an application, we study the scattering of quaternionic particles through a scalar step potential. We also provide a general solution method for the quaternionic Schrödinger equation, which can be applied to more sophisticated and physically interesting models.
We extend Pearl's definition of causal influence to the quantum domain, where two quantum systems $A$, $B$ with finite-dimensional Hilbert space are embedded in a common environment $C$ and propagated with a joint unitary $U$. For finite dimensional Hilbert space of $C$, we find the necessary and sufficient condition on $U$ for a causal influence of $A$ on $B$ and vice versa. We introduce an easily computable measure of the causal influence and use it to study the causal influence of different quantum gates, its mutuality, and quantum superpositions of different causal orders. For two two-leve…
The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics
本文的主要目的是对相对论性量子力学进行一种本质上的概率分析。它是基于这样的假设，即每当概率分布出现时，存在一个随机过程，该过程要么负责给定测度的时间演化，要么在平稳情况下保留该测度。我们的出发点是所谓的概率演化的薛定谔问题，它提供了一个唯一的马尔可夫随机插值之间的任何给定的边界概率密度对一个过程，覆盖一个固定的，有限的持续时间，提供了我们已经决定了先验什么样的原始动力学半群过渡机制参与。在非相对论理论中，包括量子力学，Feyman-Kac-like内核是构建合适的跃迁概率密度的过程。在标准的“自由”情况下（费曼-卡茨势等于零），熟悉的维纳噪声恢复。在薛定谔问题的框架中，“自由噪声”也可以扩展到任何无限可分的概率定律，如列维-欣钦公式所涵盖的。由于相对论哈密顿量$|\nabla |$和$\sqrt {-\triangle +m^2}-m$已知产生这样的法律，我们专注于他们的概率现象的分析，这是与相对论波（达朗贝尔）和物质波（克莱因-戈登）方程，分别。我们证明了这样的随机过程存在，是空间跳跃过程。在一般情况下，在存在的外部潜力，他们不共享马尔可夫性质，除了平稳的情况。详细分析了伪微分柯西-薛定谔演化的一个具体例子。相关波的相对论协方差
Unusual scalar products in Hilbert space of Quantum Mechanics: non-Hermitian square-well model with two coupled channels
提出了耦合沟道方势阱的赝厄米模型，并对其进行了求解和讨论。参数的域是在所有的束缚态能量（相对于第二个可观察的，我们称之为“自旋”的两次退化）保持真实的地方确定的。波函数的标准概率解释是通过从原始的基本非物理不定伪度量$θ$过渡到另一个更复杂但正确的正定物理度量$Θ$来实现的。在我们的模型中，后一步仍然相对容易可行。在微扰构造度量$Θ$的过程中，出现了另一种幸运的情况，因此，也就是构造正确的标积和物理范数的过程。我们发现，在较高的能级上，我们模型的所有非厄米性的影响变得非常强烈，并逐渐受到抑制。
Harmonizing de Broglie-Bohm's Causal Interpretation with the Copenhagen Interpretation of Quantum Mechanics
本文通过引入动量非线性算符，调和德布罗意-玻姆因果诠释与哥本哈根诠释，并证明该算符隐含于前者理论中。
The laws of quantum mechanics are couched in subtle mathematical language. The laws are not usually stated in a compact pedagogical form. Here I present a possible way to correct this. Essential facts can be distilled into seven statements that are easy to remember and easily referred back. Also, the current teaching of quantum mechanics is laden with words of negative connotations, originating as they did during the early decades of the subject when the subject was intellectually still puzzling. A wide variety of experiments in the intervening decades, not least those that were awarded the No…
Popper's experiment, Copenhagen Interpretation and Nonlocality
本文对卡尔·波普尔提出的一个最近在实验上实现的思想实验进行了批判性的考察。波普尔的论点，也一直在随后的辩论中盛行的一个基本缺陷，指出。结果表明，波普尔实验可以很容易地理解在哥本哈根解释的量子力学。提出了一个基于离散变量的替代实验，它以更清晰的方式构成了波普尔检验。反驳了量子力学中不存在非定域性的论点。
An approach to construct wave packets with complete classical-quantum correspondence in non-relativistic quantum mechanics
介绍了一种在一维非相对论量子力学中构造具有完全经典对应和量子对应的波包的方法。首先，我们考虑两个总能量相等的相似振子。在经典域中，我们可以很容易地求解这个模型，得到变量空间中的轨迹。量子水平上的这幅图等价于一个双曲型偏微分方程，它给了我们选择初始波函数及其初始斜率的自由。利用这一自由度，我们提出了一种方法来选择一个适当的初始条件，这是独立于振子的形式。然后，我们构造了某些情况下的波包，并表明这些波包密切遵循整个经典轨迹和峰值上。此外，我们使用德布罗意玻姆解释的量子力学量化这种对应关系，并表明，由此产生的玻姆轨迹也是在一个完全的协议与他们的经典同行。
We present an alternative formalism of quantum mechanics tailored to statistical ensemble in phase space. The purpose of our work is to show that it is possible to establish an alternative autonomous formalism of quantum mechanics in phase space using statistical methodology. The adopted perspective leads to obtaining within the framework of its theory the master quantum-mechanical equation without recourse to the other formalisms of quantum mechanics, and gives the idea of operators pertaining to dynamical quantities. The derivation of this equation starts with the ensemble in phase space. We…
We examine a deformed quantum mechanics in which the commutator between coordinates and momenta is a function of momenta. The Jacobi identity constraint on a two-parameter class of such modified commutation relations (MCR's) shows that they encode an intrinsic maximum momentum; a sub-class of which also imply a minimum position uncertainty. Maximum momentum causes the bound state spectrum of the one-dimensional harmonic oscillator to terminate at finite energy, whereby classical characteristics are observed for the studied cases. We then use a semi-classical analysis to discuss general concave…
Ockham's razor is a heuristic concept applied in philosophy of science to decide between two or more feasible physical theories. Ockham's razor operates by deciding in favour of the theory with least assumptions and concepts; roughly speaking, the less complex theory. Could Ockham's razor not easily treat the different interpretations as theories and decide in favour of the one with fewest assumptions? We provide an answer to this question by means of examples of applications in literature and the discussion of its historical origin. We review the historical context of Ockham's razor and its c…
A Fisher Information Perspective of Relativistic Quantum Mechanics
该研究从费希尔信息视角出发，将此前基于流体动力学拉格朗日量推导非相对论量子力学方程的方法拓展至相对论流，为推导相对论量子力学提供了新的研究路径。
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