On some peculiarities of quantum mechanics

General regularities related toLagrangian andHamiltonian equations are revealed. Probability distributions for functions ofHamiltonian random variables are considered. It is shown that all probability distributions of this kind are fully determined by the probability distributions for the random variables satisfying the corresponding Lagrangian equations. Some formulas related tocanonically conjugate operators are given. The similarity of these formulas to those related to Hamiltonian random variables is demonstrated. The quantum approach to the treatment of Hamiltonian random variables is discussed, and the origin of some peculiarities related to this approach is elucidated; it is explained, in particular, why it is impossible to form the joint probability density for canonically conjugate random variables when using this approach. The peculiarities revealed prove to be common for any objects possessing Hamiltonian random variables, irrespective of the nature of the objects, and coincide, therefore, with those in quantum mechanics. The existence of joint probability distributions for canonically conjugate random variables in the general case is demonstrated through the calculation of the corresponding joint mathematical expectations in an illustrative example. This proves, in particular, that joint probability distributions for canonically conjugate coordinates and momenta exist indeed in the case of mechanical microsystems. The results obtained prove once again that the pecularities of quantum mechanics are not related to the specificity of the measurements of physical quantities for microsystems.     

On some hydrodynamical aspects of quantum mechanics

In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the complex polynomial (i.e. Borel-Weil) realization of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dimensional geometric quantum mechanics, the Schrödinger velocity field on projective Hilbert space is divergence-free (being Killing with respect to the Fubini-Study metric) and fulfils the stationary Euler equation, with pressure proportional to the Hamiltonian uncertainty (squared). We explicitly determine the critical points of the pressure of this “Schrödinger fluid”, together with its vorticity, which turns out to depend on the spacings of the energy levels. These results follow from hydrodynamical properties of Killing vector fields valid in any (finite dimensional) Riemannian manifold, of possible independent interest.     

Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

On convenient parametrization of boundary conditions for some models of quantum mechanics

On the example of some simple models of quantum mechanics we present a convenient way of parametrization of the most general boundary condition. In particular, the proposed parametrization allows investigating the influence on the energy spectra of boundary conditions, which break the natural symmetries of the Schrodinger equation. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Monte-carlo calculations for some problems of quantum mechanics

The Monte-Carlo technique for the calculations of functional integral in two one-dimensional quantum-mechanical problems had been applied. The energies of the bound states in some potential wells were obtained using this method. Also some peculiarities in the calculation of the kinetic energy in the ground state had been studied.     

Measurement problem in quantum mechanics: Characteristics of an apparatus

The problem of measurement in Quantum Mechanics will be briefly reviewed. Since the measurement process involves a macroscopic apparatus, the attention is focussed on the dynamics of a pointer-like variable of the apparatus when it interacts with a quantum system. It is argued that since the measurement process requires an apparent collapse of the wave function in a certain basis, and collapse is an irreversible process, understanding of irreversibility in a quantum macroscopic system is crucial. The chief characteristics of an apparatus that are important in understanding measurement process are (a) its closely spaced energy levels and (b) its interaction with environment. The coupling with the environment drives the density matrix of the apparatus to diagonal form, but to have persistent correlations between system and apparatus states, it seems necessary to have a pointer variable that has a classical limit     

Remote two-time boundary conditions and special states in quantum mechanics

For an oscillating universe, the assumptions of wave-function localization and approximate symmetry for the initial and final states impose restrictions on the intermediate motion. We argue that one of these restrictions is the avoidance of superpositions of macroscopically distinct states.     

Toward a superconducting quantum computer: Harnessing macroscopic quantum coherence

Intensive research on the construction of superconducting quantum computers has produced numerous important achievements. The quantum bit (qubit), based on the Josephson junction, is at the heart of this research. This macroscopic system has the ability to control quantum coherence. This article reviews the current state of quantum computing as well as its history, and discusses its future. Although progress has been rapid, the field remains beset with unsolved issues, and there are still many new research opportunities open to physicists and engineers.     

Enhancement of feasibility of macroscopic quantum superposition state with the quantum Rabi-Stark model

We propose an efficient scheme to generate a macroscopical quantum superposition state with a cavity optomechanical system, which is composed of a quantum Rabi-Stark model coupling to a mechanical oscillator. In a low-energy subspace of the Rabi-Stark model, the dressed states and then the effective Hamiltonian of the system are given. Due to the coupling of the mechanical oscillator and the atom-cavity system, if the initial state of the atom-cavity system is one of the dressed states, the mechanical oscillator will evolve into a corresponding coherent state. Thus, if the initial state of the atom-cavity system is a superposition of two dressed states, a coherent state superposition of the mechanical oscillator can be generated. The quantum coherence and their distinguishable properties of the two coherent states are exhibited by Wigner distribution. We show that the Stark term can enhance significantly the feasibility and quantum coherence of the generated macroscopic quantum superposition state of the oscillator.     

量子力学诠释问题

量子力学的建立不仅奠定了当代科学的基础,而且在推动当代技术革命方面取得了惊人的成功。然而,对于量子力学诠释(interpretation of quantum mechanics)——理解波函数如何刻画微观世界,人们迄今为止并未形成共识。量子力学发展的这种二元状态不仅带来了认识论方面的误导,而且依据备受争议的哥本哈根诠释建立起来的量子技术会有许多根本性问题。
量子力学的哥本哈根诠释存在二元结构的问题：微观世界的运动用量子力学描述,是一个幺正演化,而观察或测量却依赖于量子系统外部的经典世界(仪器、观察者、环境),表现出来的波包塌缩是非幺正的。为此,包括爱因斯坦、薛定谔、温伯格等在内的一些著名学者对哥本哈根诠释提出了尖锐的批评。80年过去了,为克服量子力学的哥本哈根诠释二元论困境,人们提出各种各样的量子力学诠释,包括多世界诠释、量子退相干诠释、自洽历史诠释以及量子达尔文主义等。文章将简要介绍和评述这些量子力学诠释的基本思想、它们之间的逻辑关系及其实验检验的可能性。进一步澄清量子力学诠释中的基本概念,可以避免量子观念滥用导致的意识论上的问题和量子技术发展误入歧途。     

稀薄原子气体玻色-爱因斯坦凝聚近期研究进展简介

论述了自1999年涡旋态在稀薄碱金属原子气体的玻色-爱因斯坦凝聚体(BEC)中成功实现以来有关的BEC理论和实验研究的进展及作近期的部分探索，并介绍了相关的基本概念和理论。     

Dynamical time versus system's time in quantum mechanics

Properties of an operator representing the dynamical time in the extended parameterization invariant formulation of quantum mechanics are studied. It is shown that this time operator is given by a positive operator measure analogously to the quantities that are known to represent various measurable time operators. The relation between the dynamical time of the extended formulation and the best known example of the system's time operator, i.e., for the free one-dimensional particle, is obtained.     

Dynamical time versus system time in quantum mechanics

Properties of an operator representing the dynamical time in the extended parameterization invariant formulation of quantum mechanics are studied. It is shown that this time operator is given by a positive operator measure analogously to the quantities that are known to represent various measurable time operators. The relation between the dynamical time of the extended formulation and the best known example of the system time operator, i.e., for the free one-dimensional particle, is obtained.     

The neutron interferometer as a macroscopic quantum device

Neutron interferometry is a unique tool for investigations in the field of particle-wave dualism because massive elementary particles behave like waves within the interferometer. The invention of perfect crystal neutron interferometers providing widely separated coherent beams stimulated a great variety of experiments with matter waves in the field of basic quantum mechanics. The phase of the spatial and spinor wave function become a measurable quantity and can be influenced individually. High degrees of coherence and high order interferences have been observed by this technique. The 4π-symmetry of a spinor wave function and the mutual modulation of nuclear and magnetic phase shifts have been measured in the past. Recent experiments dealt with polarized neutron beams, which are handled to realize the spin-superposition of two oppositionally polarized subbeams resulting in a final polarization perpendicular to both initial beam polarizations. The different actions on the coherent beams of static (DC) and dynamic (HF) flippers have been visualized.     

On the geometrization of quantum mechanics

Nonrelativistic quantum mechanics is commonly formulated in terms of wavefunctions (probability amplitudes) obeying the static and the time-dependent Schrödinger equations (SE). Despite the success of this representation of the quantum world a wave–particle duality concept is required to reconcile the theory with observations (experimental measurements). A first solution to this dichotomy was introduced in the de Broglie–Bohm theory according to which a pilot-wave (solution of the SE) is guiding the evolution of particle trajectories. Here, I propose a geometrization of quantum mechanics that describes the time evolution of particles as geodesic lines in a curved space, whose curvature is induced by the quantum potential. This formulation allows therefore the incorporation of all quantum effects into the geometry of space–time, as it is the case for gravitation in the general relativity.     

统计物理的微观动力学基础

统计的基本出发点是研究系统具有的随机性,不同系统在不同情形下的宏观热力学性质起源于系统内部随机性的差异,通过对宏观热力学系统的微观非线性动力学进行研究探索，我们可以进一步更为深入地理解物态方程、相变等诸多的宏观热力学现象。本文通过哈密顿系统的非线性动力学研究，以及遍历性理论的动力学随机性研究对此问题进行了分析，研究表明，动力学系统的全局性混沌是系统统计成立的根本要素，系统的无限大自由度（热力学极限）已不是决定性的因素，人们可以在此基础上建立少自由度系统的统计力学及热力学。