A New View of Effects in a Hilbert Space

Paul Busch, Stanley Gudder

Robert F. O’Connell, A. K. Rajagopal

A new interpretation of the scalar product in Hilbert space was presented recently by Aharonov, Albert, and Au. Here the essential equivalence between such a view-point and Wigner's exact reformulation of nonrelativistic quantum mechanics using distribution functions is pointed out.

Timothy J. Moreland, Stanley Gudder

David Carfì

In this paper we study from a topological point of viewthe Hellmann-Feynman theorem of Quantum Mechanics. The goal of the paper is twofold: On one hand we emphasize the role of the strong topology in the classic version of the theorem in Hilbert spaces, for what concerns the kind of convergence required on the space of continuous linear endomorphisms, which contains the space of (continuous) observables. On the other hand we state and prove a new pointwise version of the classic Hellmann-Feynman theorem. This new version is not yet present in the literature and follows the idea of A. Bohm concerning the topology which is desiderable to use in Quantum Mechanics. It is indeed out of question that this non-trivial new version of the Hellmann-Feynman theorem is the ideal one - for what concerns the continuous observables on Hilbert spaces, both from a theoretical point of view, since it is the strongest version obtainable in this context - we recall that the pointwise topology is the coarsest one compatible with the linear structure of the space of continuous observables -, and from a practical point of view, because the pointwise topology is the easiest to use among topologies: it brings back the problems to the Hilbert space topology. Moreover, we desire to remark that this basic theorem of Quantum Mechanics, in his most desiderable form, is deeply interlaced with two cornerstones of Functional Analysis: the Banach-Steinhaus theorem and the Baire theorem.

K. Kraus

Lu Dao

Using the numerical method,the antibunching effects of a new kind of odd and even nonlinear coherent states in a finite-dimensional Hilbert space are studied.The results show that the new kind of odd nonlinear coherent state exhibits antibunching effect and the new kind of even nonlinear coherent state does not exhibits antibunching effect in a finite-dimensional Hilbert space.