Asymptotic localization and separation and the scattering of quantum states

The concepts of asymptotic localization and separation of quantum states introduced in a previous paper are generalized to the n-dimensional case. An application of these ideas to a simple scattering situation leads to an intuitively pleasing picture of scattering.     

On the equilibrium states in quantum statistical mechanics

Representations of theC*-algebra of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential are studied. Both for finite and for infinite systems it is shown that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of onto its commutant. This means that there is an equivalent anti-linear representation of in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.     

On locally normal states in quantum statistical mechanics

A sufficient condition is given in order that a von Neumann algebra with cyclic vector is quasi-standard. With the help of this result it is proved that a locally normal state with a cyclic and separating vector in the representation space gives rise to a quasi-standard von Neumann algebra. Furthermore it is proved that the representation space determined by a locally normal state in the G.N.S. construction is separable.     

Naturalness of the space of states in quantum mechanics

We show how certain constructions of quantum mechanics, like monopoles, instantons, and the Schrödinger-von Neumann equation, are related to geometric functors which are representable. We study the differential geometry of the projective bundle associated with an infinite-dimensional separable Hilbert space, and we construct a universal connection which, is described as a subspace of skwe-Hermitian operators. This connection is responsible for the Berry phase.     

谈谈量子力学中的状态叠加原理

以对话的形式,介绍并评论了布洛欣采夫、狄拉克以及朗道和栗弗席茨关于状态叠加原理的不同表述.     

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.     

Stochastic differential equations anda posteriori states in quantum mechanics

In recent years a consistent theory describing measurements continuous in time in quantum mechanics has been developed. The result of such a measurement is atrajectoryfor one or more quantities observed with continuity in time. Applications are connected especially with detection theory in quantum optics. In such a theory of continuous measurements one can ask what is the state of the system given that a certain trajectory up to timet has been observed. The response to this question is the notion ofa posteriori states and afilteringequation governing the evolution of such states: this turns out to be a nonlinear stochastic differential equation for density matrices or for pure vectors. The driving noise appearing in such an equation is not an external one, but its probability law is determined by the system itself (it is the probability measure on the trajectory space given by the theory of continuous measurements).     

Spin and quark states in the probability representation of quantum mechanics

A short review of the general principles of constructing tomograms of spin and quark states is presented.     

关于量子力学中态叠加原理的讨论

讨论了量子力学中态叠加原理的意义,评论了它的不同表述和解说,并且以简单明了的语言来叙述这一原理．     

Extended SUSY quantum mechanics, intertwining operators and coherent states

We propose an extension of supersymmetric quantum mechanics which produces a family of isospectral Hamiltonians. Our procedure slightly extends the idea of intertwining operators. Several examples of the construction are given. Further, we show how to build up vector coherent states of the Gazeau-Klauder type associated to our Hamiltonians.     

On the definition and evolution of states in relativistic classical and quantum mechanics

Some of the problems associated with the construction of a manifestly covariant relativistic quantum theory are discussed. A resolution of this problem is given in terms of the off mass shell classical and quantum mechanics of Stueckelberg, Horwitz and Piron. This theory contains many questions of interpretation, reaching deeply into the notions of time, localizability and causality. A proper generalization of the Maxwell theory of electromagnetic interaction, required for the well-posed formulation of dynamical problems of systems with electromagnetic interaction is discussed, and some of the significance of recently found (classical) relativistic chaotic behavior is pointed out. Many results of a technical nature have been achieved in this framework; in this paper, some of these are reviewed, but I shall concentrate on a discussion of the basic ideas and foundations of the theory.On sabbatical leave from School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel.     

On phase-space representations of quantum mechanics using Glauber coherent states

A phase-space formulation of quantum mechanics is proposed by constructing two representations (identified as pq and qp) in terms of the Glauber coherent states, in which phase-space wave functions (probability amplitudes) play the central role, and position q and momentum p are treated on equal footing. After finding some basic properties of the pq and qp wave functions, the quantum operators in phase-space are represented by differential operators, and the Schrödinger equation is formulated in both pictures. Afterwards, the method is generalized to work with the density operator by converting the quantum Liouville equation into pq and qp equations of motion for two-point functions in phase-space. A coordinate transformation between those points allows one to construct a cell in phase-space, whose central point can be treated as a parameter. In this way, one gets equations of motion describing the evolution of one-point functions in phase-space. Finally, it is shown that some quantities obtained in this paper are related in a natural way with cross-Wigner functions, which are constructed with either the position or the momentum wave functions.     

Bound states in continuum of complex potentials generated by supersymmetric quantum mechanics

It is shown that bound states in continuum can be generated by applying supersymmetric quantum mechanics to real potentials in such a way that the transformed potential becomes complex. Three cases of the initial potential are considered in more detail: flat potential, step potential and rectangular quantum well. The conditions are found for the existence of bound state in continuum in these cases, as well as the expressions for the wavefunction of this state and the corresponding complex potential in analytic form.     

Electronic spectrum and localization of electronic states in aperiodic quantum dot chains

The electronic energy spectra of aperiodic Thue-Morse, Rudin-Shapiro, and double-periodic quantum dot chains are investigated in the tight-binding approximation. The dependence of the spectrum on all parameters of a “mixed” aperiodic chain model is studied: the electronic energy at quantum dots and the hopping integrals. The electronic degree of localization in the chains under consideration is determined by analyzing the inverse participation ratio. Its spectral distribution and the dependence of the band-averaged degree of localization on these model parameters have been calculated. It is shown that a transition of the system’s sites to a resonant state in which the degree of electron localization decreases, while an overlap between the subbands occurs in the spectrum is possible when the parameters are varied.     

The superposition of the states and the logic approach to quantum mechanics

An axiomatic approach to quantum mechanics is proposed in terms of a logic scheme satisfying a suitable set of axioms. In this context the notion of pure, maximal, and characteristic state as well as the superposition relation and the superposition principle for the states are studied. The role the superposition relation plays in the reversible and in the irreversible dynamics is investigated and its connection with the tensor product is studied. Throughout the paper, theW ^*-algebra model, which satisfies our axioms, is used to exemplify results and properties of the general scheme.     

Asymptotic ratio set of von neumann algebras generated by temperature states in statistical mechanics

We prove that the von Neumann algebras generated by temperature states in Statistical Mechanics have the positive real axis as their asymptotic ratio set.