A characterization of the classical states of the quantum harmonic oscillator by means of de Finetti's theorem

Integral representations of statistical operators in terms of coherent states are derived by means of a quantum version of the Poisson limit of de Finetti's theorem.     

Supersymmetric quantum mechanics and the index theorem

We consider the classical mechanics of the spinning particle and investigate which Abelian interactions can be added without breaking supersymmetry. A quantum theory is presented. The well known index theorem for the Dirac operator is extended to take into account the effect of anti-symmetric Abelian tensor fields. Furthermore interactions with non-Abelian anti-symmetric tensor fields are investigated. It turns out in both cases that these fields do not give any non-trivial contributions to the index.     

Hidden variables and Bell's theorem in quantum mechanics

In the present paper we give a precise definition of a hidden-variable theory for quantum mechanics, whereby we adopt the weakest possible definition of a hidden-variable theory, which is compatible with the assumption that the bounded observables of a quantum mechanical system are represented by the elements of the real part A_r of a W^*-algebra A (of the most general type) and the states are represented by the normal states (in the mathematical sense) of A. We then go on to show that an example put forward by Bell in 1966 satisfies our definition (Sec. 2). Finally we make use of Bell's famous theorem to show that for a sufficiently non-commutative W^*-algebra A no hidden-variable theory in our sense exists (Theorem 3.3 and its corollaries).     

Locality,Bell's theorem,and quantum mechanics

Classical relativistic physics assumes that spatially separated events cannot influence one another (locality) and that values may be assigned to quantities independently of whether or not they are actually measured (realism). These assumptions have consequences—the Bell inequalities—that are sometimes in disagreement with experiment and with the predictions of quantum mechanics. It has been argued that, even if realism is not assumed, the violation of the Bell inequalities implies nonlocality—and hence that radical changes are necessary in the foundations of physics. We show that this conclusion does not follow unless the locality hypothesis is strengthened in an implausible manner.     

Supersymmetry in problems of quantum mechanics

A connection is discussed between the group SU(2) and supersymmetry for a series of quantum mechanical problems. It is pointed out that the impossibility of factorizing Hamiltonians obtained based on representations of the group SU(2) indicates that the supersymmetry of the system is broken.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 19–23, July, 1988.In conclusion, we thank A. V. Borisov, V. Ch. Zhukovskii, I. V. Tyutin, and S. M. Khoroshkin for discussing results of this work.     

Kochen-Specker theorem in the modal interpretation of quantum mechanics

According to the modal interpretation of quantum mechanics, subsystems of a quantum mechanical system have definite properties, the set of definite properties forming a partial Boolean algebra. It is shown that these partial Boolean algebras have no common extension (as a partial Boolean subalgebra of the properties of the total system) that is embeddable in a Boolean algebra. One has thus either to restrict the rules to preferred subsystems (Healey), or to advocate a shift in metaphysics (Dieks).     

Controversial problems of measurements within quantum mechanics

It is explained why quantum mechanics applies principally to single systems and not to ensembles. A thorough analysis of thought-experiments shows clearly that irreversibility is connected with the storing of information rather than with the act of measurement. In order to avoid paradoxes one has to admit that the wave function does not represent the state of the system in itself, but information acquired in consequence of a complete measurement. The meaning of the time-energy uncertainty relation for stable systems is discussed.     

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.     

Extension of the electrokinematics theorem to the electromagnetic field and quantum mechanics

Summary A recent electrokinematics theorem leads to a general equation that, through an arbitrary irrotational fieldF, connects the motion of the electric-charge carriers, the internal potential and the dielectric properties of a physical system with its external currents, voltages and powers. It has been proved for quasi-electrostatic fields,i.e. when the vector potential may be disregarded, and on the basis of classical mechanics. Here the theorem is extended to any type of electromagnetic field and to quasi-relativistic quantum mechanics, in the case of many-particle systems for which, moreover, the probability current density is suitably computed. The new equation so obtained, throughF, connects the external currents again with the internal electric permittivity and the scalar potential, in the same way as in the preceding approach, and with the carrier velocity that, however, has to be computed according to quantum mechanics. Moreover, it contains two new contributions, one deriving from the vector potential and the other from a current density arising from the electron spin. By means of proper choices ofF, new expressions of the external currents of the system are determined as functions of the motion of its internal carriers. In particular, the electrokinematics theorem is exploited to compute the output current in two-terminal nanoelectronic devices in which, owing to the small sizes, quantum effects cannot be disregarded. Finally, such results, when they are applied to the double-barrier tunnelling structures, allow us to show the splitting of the electron pulse into two uncorrelated pulses, and as a consequence, to obtain a possible shot noise suppression, up to fifty per cent of the full shot noise.     

几个量子力学问题的统一解法

通过变换求得连带Laguerre方程的变通形式，它包括了量子力学中-维谐振子、三维谐振子和类氢离子所满足的径向薛定谔方程，因此可用统一的方法对这三个问题的能级和波函数求解。     

Comments on “locality,Bell's theorem,and quantum mechanics”

Two different ideas of locality are described. Both are due essentially to einstein. Quantum theory is compatible with the first but not the second. The problems encountered in the article cited in the title arise from trying to use only the first idea of locality, whereas Bell's-theorem considerations pertain to the second.     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’     

Quasiaccurately solvable quantum mechanics problems and the anharmonic oscillator problem

The basic methods of obtaining the anharmonic oscillator spectrum are briefly reviewed. An approach is discussed to this problem from the point of view of one of the quasiaccurately solvable models of quantum mechanics, whose wave functions are applicable to the construction of a regular perturbation theory. Results are presented of numerical calculations of the ground state energy. Analytic calculations are provided of integrals of the form , occurring in applications of the method of quasiaccurately solvable problems. M. V. Lomonosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 76–88, February, 1993.     

Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics

In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present two proofs of this theorem which are both elementary and economical. Central to our proofs is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis often focuses on the behaviour of certain two-dimensional subspaces of the Hilbert space under the action of a given Wigner symmetry, but the relevance of this behaviour to the larger picture of the whole Hilbert space is made transparent at every stage.     

An experiment to distinguish between de Broglie-Bohm and standard quantum mechanics

An experiment is suggested that is capable of distinguishing between the de Broglie-Bohm theory and standard quantum mechanics.     

Application of group symmetry to spectral problems of quantum mechanics

On the basis of a representation of the SU(2) group, a multiparameter potential of the anharmonic oscillator is constructed. A method of finding its solutions is indicated, and in particular cases, explicit expressions for the wave function and the corresponding energy values are obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No.5, pp. 90–94, May, 1988.In conclusion, the authors thank A. V. Borisov and V. Ch. Zhukovskii for useful discussion of some results of this paper.     

On the incompatibility of standard quantum mechanics and conventional de Broglie-Bohm theory

It is shown that conventional de Broglie-Bohm quantum theory is incompatible with the standard quantum theory of a system unless the former is ergodic.     

The transitions among classical mechanics,quantum mechanics,and stochastic quantum mechanics

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.     

The Atiyah-Singer index theorem as passage to the classical limit in quantum mechanics

A new approach to the Atiyah-Singer index theorem is described, using the technique of continuous fields ofC ^*-algebras. The proof is given in the case of elliptic pseudodifferential operators on ℝ ⁿ .     

Some problems of quantum mechanics possessing a non-negative phase-space distribution function

In order to clarify physical consequences due to the presence of a set of auxiliary functions _k (q,t) in quantum mechanics with a non-negative phase-space distribution function, the simplest quantum-mechanical problems are solved. It is shown that _k (q,t) influence upon the results of a problem. Therefore it is supposed that _k (q, t) reflect some physical reality (subquantum situation), interacting with a mechanical system. In particular the subquantum situation determines the minimum coordinate and momentum uncertainties ((q)² and (p)²) as well as the coordinate distribution of a fixed system and the momentum distribution of a free system. These results provide the opportunity to formulate the notion of a stationary homogeneous isotropic subquantum situation. Supposing thatq andp are small an attempt is made to develop an approximate method of solutions (quasi-orthodox approximation). Energy spectrum of an electron in a hydrogen atom is found in the second order of this approximation.On leave of absence from Peoples' Friendship University, Chair of Theoretical Physics, 3, Ordjonikidze Street, B-302, Moscow, U.S.S.R.