Brownian local time and quantum mechanics

LetV be any (sufficiently regular) attractive potential in one and two dimensions. We make rigorous an argument of M. Kac [1], relating the recurrence of the Brownian motion to the existence of at least one bound state for the quantum HamiltonianH = –(/2)+ V.     

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.     

Bounds on exponentials of local number operators in quantum statistical mechanics

We consider the quantum systems of interacting Bose particles confined to a bounded region of the configuration spaces ^v . For a class of superstable interactions we obtain bounds on exponentials of local number operators for any temperature and activity. The method we use is the Wiener integral formalism in statistical mechanics. As a consequence any thermodynamic limit states are entire analytic and locally normal in the CCR algebra. In some cases these are modular states.Research supported in part by a grant from Korean Science Foundation     

Quaternionic quantum mechanics is consistent with complex quantum mechanics

Quaternionic quantum mechanics is investigated in the light of the great success of complex quantum mechanics. It is shown that to reproduce the results of complex quantum mechanics, quaternionic quantum mechanics must contain complex quantum mechanics.     

The compatibility between quantum mechanics and local realist theories in atomic cascade experiments

We show that the divergence between the predictions of quantum optics and the local realist theory known as stochastic optics, for the extended type of photon-coincidence experiment described recently by de Caro, is of the same order of magnitude as for Aspect-type experiments. This means that, in such new experiments, as in those so far performed, counting statistics will have to be greatly improved before a discrimination between the two theories becomes possible.We also show that the outstanding difference between the two theories is that, while stochastic optics uses a genuine, that is, positive, detection probability, the corresponding quantum formalism leads to a pseudoprobability. Nevertheless, there is a striking parallel, in that both theories recognize the phenomenon known as enhancement, and both describe it as having its origin in a mixing of the signal field with the zero-point field by a polarizing device.     

Unitary state preparation,local position measurements,and spin in quantum mechanics

The orthodox presentation of quantum theory often includes statements on state preparation and measurements without mentioning how these processes can be achieved. The often quoted projection postulate is regarded by many as problematical. This paper presents a systematic framework for state preparation and measurement. Within the existing Hilbert space formulation of quantum mechanics for spinless particles we show that it is possible (1)to prepare an arbitrary state and (2)to reduce all quantum measurements to local position measurements in an asymptotic way by unitary evolution processes without recourse to the projection postulate. A generalization to spin-1/2particles is also given. The theory presented provides a general mathematical and theoretical foundation for many practical schemes for state preparation and measurement.     

Modalities and quantum mechanics

Three approaches concerning the usage of modalities in the language of quantum mechanics were considered; Mittelstaedt and I built up a dialog semantics for modalities on a metalinguistic level, and a calculus of quantum modal logic is known that is complete and sound with respect to this dialogic semantics. Van Fraassen replaced the usual interpretation of quantum mechanics (with the projection postulate) by his modal interpretation based on a modal object language. Dalla Chiara translated a nonmodal object language for quantum mechanics and the appropriate quantum logic into a modal language. Specifically we are interested in the similarities and the differences of these three approaches.     

Fractals and quantum mechanics

A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics.     

Causality and quantum mechanics

Statistical causality is recommended as the name of the generalized causality needed in quantum mechanics, instead of statistical correspondence used by Pauli.     

Relativity and quantum mechanics

Conditions under which quantum mechanics can be made compatible with the curved space-time of gravitation theories is investigated. A postulate is imposed in the formv=v _g wherev is the kinematical Hamilton-Jacobi (geometric optic limit) velocity andv _g is the group velocity of the waves. This imposes a severe condition on the possible coordinates in which the Schrödinger form (the coordinate realization) of quantum mechanics can be set up for purposes of calculating observable effects. Some such effects are calculated for a class of theories and are compared with experiments.     

Analysis of a recent experiment seemingly favouring local realism against quantum mechanics

A recent experiment by Brida et al. [Eur. Phys. J. D 44, 547 (2007)] is analyzed with the conclusion that the results disagree with standard quantum predictions but fit a simple local hidden variables model. New experiments are proposed which might throw new light on the anomaly.     

Copenhagen quantum mechanics

In our quantum mechanics courses, measurement is usually taught in passing, as an ad-hoc procedure involving the ugly collapse of the wave function. No wonder we search for more satisfying alternatives to the Copenhagen interpretation. But this overlooks the fact that the approach fits very well with modern measurement theory with its notions of the conditioned state and quantum trajectory. In addition, what we know of as the Copenhagen interpretation is a later 1950s development and some of the earlier pioneers like Bohr did not talk of wave function collapse. In fact, if one takes these earlier ideas and mixes them with later insights of decoherence, a much more satisfying version of Copenhagen quantum mechanics emerges, one for which the collapse of the wave function is seen to be a harmless book keeping device. Along the way, we explain why chaotic systems lead to wave functions that spread out quickly on macroscopic scales implying that Schrödinger cat states are the norm rather than curiosities generated in physicists’ laboratories. We then describe how the conditioned state of a quantum system depends crucially on how the system is monitored illustrating this with the example of a decaying atom monitored with a time of arrival photon detector, leading to Bohr’s quantum jumps. On the other hand, other kinds of detection lead to much smoother behaviour, providing yet another example of complementarity. Finally we explain how classical behaviour emerges, including classical mechanics but also thermodynamics.     

Symplectic quantum mechanics

Using the notion of symplectic structure and Weyl (or star) product of non-commutative geometry, we construct unitary representations for the Galilei group and show how to rewrite the Schrödinger equation in phase space. This approach gives rise to a new procedure to derive Wigner functions without the use of the Liouville-von Neumann equation. Applications are presented by deriving the states of linear and nonlinear oscillators in terms of amplitudes of probability in phase space. The notion of coherent states is also discussed in this context.