Symmetric transition probabilities in convex model of quantum mechanics

The relation between the geometry and the statistics of the quantum theory is most explicit in the convex models of quantum mechanics. A class of models is investigated where the manifold S of all states is a convex set with a smooth boundary. It is shown that for these models the simple assumption about symmetry of the transition probability implies an ellipsoidal shape of S, thus leading to the “spherical” geometries described in [7].     

Quantum mechanics without wave functions

The phase space formulation of quantum mechanics is based on the use of quasidistribution functions. This technique was pioneered by Wigner, whose distribution function is perhaps the most commonly used of the large variety that we find discussed in the literature. Here we address the question of how one can obtain distribution functions and hence do quantum mechanics without the use of wave functions.     

Nonspreading wave packets in quantum mechanics

In this paper a nonspreading, unnormalizable wave packet satisfying the Schrödinger equation is constructed. A modification of the Schrödinger equation is considered which allows the normalization of the wave packet. The case is generalized for relativistic mechanics.     

Joint probabilities of noncommuting operators and incompleteness of quantum mechanics

We use joint probabilities to analyze the EPR argument in the Bohm's example of spins.⁽¹⁾ The properties of distribution functions for two, three, or more noncommuting spin components are explicitly studied and their limitations are pointed out. Within the statistical ensemble interpretation of quantum theory (where only statements about repeated events can be made), the incompleteness of quantum theory does not follow, as the consistent use of joint probabilities shows. This does not exclude a completion of quantum mechanics, going beyond it, by a more general theory of single events, using hidden variables, for example.     

Is quantum mechanics compatible with a deterministic universe? Two interpretations of quantum probabilities

Two problems will be considered: the question of hidden parameters and the problem of Kolmogorovity of quantum probabilities. Both of them will be analyzed from the point of view of two distinct understandings of quantum mechanical probabilities. Our analysis will be focused, as a particular example, on the Aspect-type EPR experiment. It will be shown that the quantum mechanical probabilities appearing in this experiment can be consistently understood as conditional probabilities without any paradoxical consequences. Therefore, nothing implies in the Aspect experiment that quantum theory is incompatible with a deterministic universe.     

Completeness of wave operators in relativistic quantum mechanics

A combination of geometric and algebraic methods is used to prove asymptotic completeness for Schrödinger-type equations with potential not vanishing at infinity along hyperboloids (in spacetime), and with the free Hamiltonian given by the (not bounded below) relativistic (mass)² operator. The proof is based on the use of a modified form of local compactness and additional geometric properties of asymptotic scattering states which are needed to distinguish them from states trapped inside some hyperboloid for all times.Supported in part by the Fund for Basic Research administered by the Israeli Academy of Sciences and Humanities Basic Research Foundation.     

Elimination of negative probabilities within the causal stochastic interpretation of quantum mechanics

Negative probabilities resulting from the Klein-Gordon equation are eliminated from quantum theory within the stochastic interpretation of quantum theory (SIQM) for spin-zero particles. The assumption of real physical paths in E₄ implies that only particles (antiparticles) of positive energies move forward in time with positive probability densities.     

Nonspreading quasi-classical wave packets in nonrelativistic quantum mechanics

A general method of constructing nonspreading quasi-classical wave packets in nonrelativistic quantum mechanics is outlined. The nonspreading wave packet for a free particle is constructed in explicit form. It is shown that, in this case, the specified accuracy of the quasi-classical solution is not lost over time.V. V. Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 94–99, June, 1994.     

Smoothed wave functions of chaotic quantum systems

It is shown that wave functions of quantum systems as ħ → 0 have an extra density near unstable periodic trajectories of the classical problem. The averaged wave function square is represented as the sum over a finite number of periodic trajectories. The contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory. The results are compared with the numerical calculations of the wave functions for the stadium billiard.     

Exact wave functions of two-electron quantum rings

We demonstrate that the Schr?dinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius. We show that both polynomial and irrational solutions can be found for any value of the angular momentum and that the singlet and triplet manifolds, which are degenerate, have distinct geometric phases. We also study the nodal structure associated with these two-electron states.     

Weak objectification,joint probabilities,and Bell inequalities in quantum mechanics

The weak objectification of physical properties is shown to yield the same probabilistic implications as strong objectification and can therefore be refuted on the basis of suitable interference experiments. An alternative test of hypothetical objectification statements, as they occur in the EPR experiment, is based on joint probabilities and the ensuing Bell inequalities. Quantum mechanics turns out to be partially compatible with Bell's inequalities even in cases where weak objectification is excluded by interference.     

Cosmology and the pilot wave interpretation of quantum mechanics

Bell has recently revived the pilot wave interpretation of de Broglie and Bohm as a possible scheme for interpreting wave functions in quantum cosmology. I argue that the pilot wave interpretation cannot be applied consistently to systems whose wave functions split into macroscopically distinguishable states. At some stage the pilot wave interpretation must either tacitly invoke wave function reduction in the same manner as the Copenhagen interpretation, or else abandon locality by requiring physical particles to move faster than light. Consequently, the many-worlds interpretation is the only known realist interpretation of the quantum mechanical formalism which can be extended to quantum cosmology.     

Bicovariant differential calculus on quantum groups and wave mechanics

The bicovariant differential calculus on quantum groups being defined by Woronowicz and later worked out explicitly by Carow-Watamura et at. and Juro for the real quantum groupsSU _q (N) andSO _q (N) through a systematic construction of the bicovariant bimodules of these quantum groups is reviewed forSU _q (2) andSO _q (N). The resulting vector fields build representations of the quantized universal enveloping algebras acting as covariant differential operators on the quantum groups and their associated quantum spaces. As an application a free particle stationary wave equation on quantum space is formulated and solved in terms of a complete set of energy eigenfunctions.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.     

The thermal Green functions in nonextensive quantum statistical mechanics

The thermal Green functions of the quantum-mechanical harmonic oscillator are constructed within the framework of nonextensive statistical mechanics with normalized q -expectation values. For the Tsallis index q greater than unity, these functions are found to be expressed analytically in terms of the Hurwitz zeta function. It is found that influence of the nonextensivity on the time-ordered thermal propagator is relevant only at the “on-shell” states. In particular, the finite-temperature contribution to the thermal propagator becomes enhanced for the strong nonextensivity. Received 30 September 1998     

Two-time correlation functions: stochastic and conventional quantum mechanics

An investigation of two-time correlation functions is reported within the framework of (i) stochastic quantum mechanics and (ii) conventional Heisenberg-Schr?dinger quantum mechanics. The spectral functions associated with the two-time electric dipole moment correlation functions are worked out in detail for the case of the hydrogen atom. While the single time averages are identical for stochastic and conventional quantum mechanics, differences arise in the two approaches for multiple time correlation functions.     

Imaging electron wave functions inside open quantum rings

Combining scanning gate microscopy (SGM) experiments and simulations, we demonstrate low temperature imaging of the electron probability density |Psi|(2)(x,y) in embedded mesoscopic quantum rings. The tip-induced conductance modulations share the same temperature dependence as the Aharonov-Bohm effect, indicating that they originate from electron wave function interferences. Simulations of both |Psi|(2)(x,y) and SGM conductance maps reproduce the main experimental observations and link fringes in SGM images to |Psi|(2)(x,y).     

Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics

We consider wave packets of free particles with a general energy-momentum dispersion relation E(p). The spreading of the wave packet is determined by the velocity v=∂_pE. The position-velocity uncertainty relation is saturated by minimal uncertainty wave packets Φ(p)=Aexp(-αE(p)+βp). In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation E(p)=p²/2m, analytic calculations are presented for the spreading of wave packets with minimal position-velocity uncertainty product for the lattice dispersion relation E(p)=-cos(pa)/ma² as well as for the relativistic dispersion relation . The boost properties of moving relativistic wave packets as well as the propagation of wave packets in an expanding Universe are also discussed.