The Phase Space Model of Nonrelativistic Quantum Mechanics

We focus on several questions arising during the modelling of quantum systems on a phase space. First, we discuss the choice of phase space and its structure. We include an interesting case of discrete phase space. Then, we introduce the respective algebras of functions containing quantum observables. We also consider the possibility of performing strict calculations and indicate cases where only formal considerations can be performed. We analyse alternative realisations of strict and formal calculi, which are determined by different kernels. Finally, two classes of Wigner functions as representations of states are investigated.     

The Projective Hilbert Space as a Classical Phase Space for Nonrelativistic Quantum Dynamics

The projective Hilbert space carries a natural symplectic structure which enables one to reformulate quantum dynamics as a classical Hamiltonian one. PACS: 03.65.Ta, 02.40.Yy, 45.20.Jj.     

An Algebra of Effects in the Formalism of Quantum Mechanics on Phase Space

Defining an addition of the effects in the formalism of quantum mechanics on phase space, we obtain a new effect algebra that is strictly contained in the effect algebra of all effects. A new property of the phase space formalism comes to light, namely that the new effect algebra does not contain any pair of noncommuting projections. In fact, in this formalism, there are no nontrivial projections at all. We illustrate this with the spin-1/2 algebra and the momentum/position algebra. Next, we equip this algebra of effects with the sequential product and get an interpretation of why certain properties fail to hold. PACS: 02.10.Gd, 03.65.Bz. This paper was a submission to the Fifth International Quantum Structure Association Conference (QS5), which took place in Cesena, Italy, March 31–April 5, 2001.     

非对易相空间中角动量的分裂

非对易空间效应是一种在弦尺度下出现的物理效应. 本文首先介绍了在Schwinger表象中角动量的3个分量用产生--消灭算符的表示形式, 接着讨论了非对易相空间的量子力学代数; 然后用对易空间谐振子的产生-消灭算符表示出了在非对易情况下的角动量; 最后讨论了非对易相空间中角动量的分裂.     

Quasiprobability Distribution and Phase Distribution in Interacting Fock Space

In this paper we study quasiprobability distribution and phase distribution for coherent states, squeezed states, and Kerr states in one-mode interacting Fock space.     

On Quantum Mechanics on Noncommutative Quantum Phase Space

In this work, we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=ε_ij^kθ_k and a momentum noncommutativity matrix parameter β=ε_ij^kβ_k, is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS). Imposing some constraintson this particular transformation, we firstly find that the product of the two parameters θ and β possesses alower bound in direct relation with Heisenberg incertitude relations, and secondly that the two parameters are equivalent but with opposite sign, up to a dimension factor depending on thephysical system under study. This means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole NCQPS. Within our framework, we treat some physical systems on NCQPS : free particle, harmonic oscillator, system of two-charged particles, Hydrogen atom. Amongthe obtained results, we discover a new phenomenon which consists of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β,representing the same particle in presence of a magnetic field$vec{B}=q^{-1}vec{beta}$. For the other examples, additionalcorrection terms depending on β appear in the expression of the energy spectrum. Finally, in the two-particle system case, we emphasize the fact that for two opposite charges noncommutativity is effectively feeled with opposite sign.     

Interference in Phase Space of Squeezed States for the Time-Dependent Hamiltonian System

We showed that the idea of Schleich and Wheeler (1987, Nature 326, 574) for the semiclassical approach of the interference in phase space of harmonic oscillator squeezed states can be extended to that of general time-dependent Hamiltonian system. The quantum phase properties of squeezed states for the general time-dependent Hamiltonian system are investigated by using the quantum distribution function. The weighted overlaps A _n and phases θ _n for the system are evaluated in the semiclassical limit.     

量子相空间纠缠轨线力学

量子相空间理论已用来研究物理学、化学等有关问题, 并为人们研究经典物理和量子物理的对应关系提供了一种有力工具. 在量子相空间中, 基于Wigner表象下的量子刘维尔方程, 建立分子纠缠轨线力学. 与经典分子力学方法不同, 分子纠缠轨线力学中的轨线不再是独立的, 而是“纠缠”在一起的, 这正是体系量子效应的体现. 这种半经典 的理论方法能给出体系的量子效应及具有启示意义的物理图像. 分子纠缠轨线力学被用来研究量子隧穿效应、分子光解反应动力学、自关联函数等. 本文综述了分子纠缠轨线力学最近的发展. 关键词： 纠缠轨线 量子相空间 半经典理论     

Algebra of Effects in the Formalism of Quantum Mechanics on Phase Space as an M. V. and a Heyting Effect Algebra

We prove that the algebra of effects in the phase space formalism of quantum mechanics forms an M. V. effect algebra and moreover a Heyting effect algebra. It contains no nontrivial projections. We equip this algebra with certain nontrivial projections by passing to the limit of the quantum expectation with respect to any density operator. PACS: Primary 02.10.Gd, 03.65.Bz, Secondary 002.20.Qs This paper was a submission to the Sixth International Quantum Structure Association Conference (QS6), which took place in Vienna, Austria, July 1–7, 2002.     

The Quantum Phase Problem: Steps Toward a Resolution

Defining the observable canonically conjugate to the number observable N has long been an open problem in quantum theory. The problem stems from the fact that N is bounded from below. In a previous work we have shown how to define the absolute phase observable || by suitably restricting the Hilbert space of x and p like variables. Here we show that also from the classical point of view, there is no rigorous definition for the phase even though it's absolute value is well defined.     

Phase Space Path Integral Method for Problem of Obtaining Propagator of a Particle with a Force Quadratic in Velocity

This article uses the phse space path integral method to find the propagator for a particle with a force quadratic in velocity. Two specific canonical transformations has been used for this purpose.     

Spin-Dependent Bohm Trajectories for Pauli and Dirac Eigenstates of Hydrogen

The de Broglie-Bohm causal theory of quantum mechanics is applied to the hydrogen atom in the fully spin-dependent and relativistic framework of the Dirac equation, and in the nonrelativistic but spin-dependent framework of the Pauli equation. Eigenstates are chosen which are simultaneous eigenstates of the energy H, total angular momentum M, and z component of the total angular momentum M _z. We find the trajectories of the electron, and show that in these eigenstates, motion is circular about the z-axis, with constant angular velocity. We compute the rates of revolution for the ground (n=1) state and the n=2 states, and show that there is agreement in the relevant cases between the Dirac and Pauli results, and with earlier results on the Schrödinger equation.     

An Approach to the Study of Quantum Phase Problem Based on Action-Angle Wigner Distribution

Quantum phase distribution is expressed in terms of action-angle Wigner distribution function. It turns out to coincide with the limit case of Pegg-Barnett theory. This discrete phase space approach, in which some concepts such as quantum phase operator are not needed, can express phase-related quantities in a unified way. The expectation values and variances of cosθ and sinθ are the same as those of Susskind-Glogolver theory. The phase-(particle) number uncertainty has a simple form in this formalism.     

Quantum nonlocality without counterfactual definiteness?

Stapp's recent proof of the nonlocality of quantum mechanics is critically discussed. It is demonstrated that in his derivation of the Bell inequalities an extra requirement, over locality, is used, which is tantamount to counterfactual definiteness, and that is not a consequence of locality.1. Research associate N.F.W.O. (Belgium).     

热真空态的Husimi分布函数

根据量子统计中相空间的理论知识,采用两种不同的方法给出了热真空态的Husimi分布函数,并给出了热真空态的Wehrl墒.最后与相应混合态的Husimi分布函数进行比较,得出来热真空态的Husimi分布函数与混合态的Husimi分布函数是相一致的结论.     

Ergodic Properties for a Quantum Nonlinear Dynamics

We present a quantum system composed of infinitely many particles, subject to a nonquadratic Hamiltonian, for which it is possible to investigate the long time behavior of the dynamics and its ergodic properties. We do so both for the KMS states and for a large class of locally normal invariant states, whose very existence is already of some interest.     

Phase-Matching Quantum Key Distribution with Discrete Phase Randomization

The twin-field quantum key distribution (TF-QKD) protocol and its variations have been proposed to overcome the linear Pirandola–Laurenza–Ottaviani–Banchi (PLOB) bound. One variation called phase-matching QKD (PM-QKD) protocol employs discrete phase randomization and the phase post-compensation technique to improve the key rate quadratically. However, the discrete phase randomization opens a loophole to threaten the actual security. In this paper, we first introduce the unambiguous state discrimination (USD) measurement and the photon-number-splitting (PNS) attack against PM-QKD with imperfect phase randomization. Then, we prove the rigorous security of decoy state PM-QKD with discrete phase randomization. Simulation results show that, considering the intrinsic bit error rate and sifting factor, there is an optimal discrete phase randomization value to guarantee security and performance. Furthermore, as the number of discrete phase randomization increases, the key rate of adopting vacuum and one decoy state approaches infinite decoy states, the key rate between discrete phase randomization and continuous phase randomization is almost the same.     

Phase Measurement in Interacting Fock Space

In this paper we discuss probability operator measure and phase measurement in one mode interacting Fock space.