The Quantum Mechanical Formalism

Nimrod Moiseyev

This book is dedicated to the non-Hermitian formalism of quantum mechanics. In this chapter we wish to give the motivation and the rational for developing a non-Hermitian formalism to quantum mechanics. Therefore this chapter will not explain how non-Hermitian calculations are carried out or in what way the non-Hermitian formalism is analogous to the standard (Hermitian) formalism of quantum mechanics. It is important to emphasize that there is no (known) transformation which enables one to map results which were obtained using one formalism to the other one. Yet, the same physical results should be obtained by studying the same phenomenon using the two formalisms. If this is the case, why should one bother to develop an alternative formalism to the standard Hermitian formalism of quantum mechanics? There are several reasons for doing this and here we shall focus on five of those reasons. (1) There are phenomena which can be explained in a straightforward fashion using the non-Hermitian formalism but are very hard and often impossible to explain within the framework of the standard (Hermitian) formalism of quantum mechanics . In particular in Chapter 9 we will describe several physical phenomena which are associated with the self-orthogonality where two or more degenerate resonance states are coalesced. (2) There are physical phenomena which one might not immediately associate with quantum behavior where the quantum language can be used to describe the physics . The studied problem may be, for example, in systems described in terms of classical statistical mechanics, diffusion in biological systems, or propagation of light in waveguides (WG). […]

Ragnar A. Hoffman, Sture Forsén, Bo Gestblom

Sol Wieder

A. Gersten, Amnon Moalem

Abstract A consistent formalism with quantum mechanics, different from Bohmian mechanics, which describes the quantum corrections to classical trajectories, is developed. The quantum potential of Bohmian mechanics is part of the kinetic energy in our formalism. The difficulties of the Bohmian mechanics formalism are pointed out.

Y. B. Band, Y. Avishai

Emmanuel E. Haven, Andrei Yu. Khrennikov

Mathematical formalism of quantum mechanics: brief introduction In this chapter, we present basic notions of quantum mechanics, and we emphasize “quantum mathematics.” To posit that the mathematical formalism of quantum mechanics can be used outside of quantum physics (as in sociology, economics, or finance) may be of interest. If one can steer away from merely positing and veer instead into convincing the reader that such a position is tenable, then some achievement has been booked. But the price to pay is that the formalism of quantum mechanics needs to be uncovered: how else can one judge whether what we posit is senseful? Thus, a reader can either (i) formally accept our viewpoint about the quantum-like paradigm (see Khrennikov [1]), or (ii) ask for a detailed analysis of the quantum foundations which would justify this paradigm. In the first case, it may well be sufficient to skip this chapter (but we should suggest to attempt to include Chapter 3). In the second case, we invite especially those readers who are concerned about the fact that the formal usage of the mathematical formalism of quantum mechanics may transfer major problems of quantum foundations to social science, economics, and finance, to read the present chapter carefully.