The present paper generalizes preceding papers of the author [1]–[6] and opens a cycle of works concerning the general quantum-mechanical inverse scattering problem (for a given partial channel) and its solution in analytic form in a nontraditional setting: the scattering wave function in the momentum representation is recovered immediately from the scattering data. The solution we find for the problem in question is given by a family of phase-equivalent wave functions with explicitly controlled functional-analytic arbitrariness. This arbitrariness corresponds to the arbitrariness in the choice of phase-equivalent nonlocal potentials and demands including some additional concepts of quantum measurement theory to a complete description of quantum-mechanical systems. We discuss several related problems of quantum-mechanical probability distributions and their statistical interpretation. The mathematical apparatus of the approach thus developed is illustrated by the example of describing the nucleon-nucleon interaction in uncoupled partial channels. In conclusion we announce the topics of forthcoming works of the series.
The present paper generalizes preceding papers of the author [1]–[6] and opens a cycle of works concerning the general quantum-mechanical inverse scattering problem (for a given partial channel) and its solution in analytic form in a nontraditional setting: the scattering wave function in the momentum representation is recovered immediately from the scattering data. The solution we find for the problem in question is given by a family of phase-equivalent wave functions with explicitly controlled functional-analytic arbitrariness. This arbitrariness corresponds to the arbitrariness in the choice of phase-equivalent nonlocal potentials and demands including some additional concepts of quantum measurement theory to a complete description of quantum-mechanical systems. We discuss several related problems of quantum-mechanical probability distributions and their statistical interpretation. The mathematical apparatus of the approach thus developed is illustrated by the example of describing the nucleon-nucleon interaction in uncoupled partial channels. In conclusion we announce the topics of forthcoming works of the series.
In order to understand nonlocal phenomena, the corresponding processes must be treated as scattering experiments and investigated in the frame of quantum eld theory. Quantum mechanics is not su cient for this task, because it is only concerned with the in uence of a potential on a physical object. But in processes with nonlocal e ects the role of interaction between physical objects must be su ciently respected. All this is shown rst of all for the special case of the spin-1/2-experiment, and then generalized to arbitrary scattering processes. 1. What is the problem? Nonlocal phenomena are discussed and analyzed, since Einstein, Podolsky and Rosen in their famous paper [1] have drawn attention to a di culty of quantum mechanics. By means of their also famous gedankenexperiment the three authors have demonstrated their thesis that quantum mechanics is incomplete. So called local theories were developed in order to repair this pretended fault. The work of Bell [2] consisted in describing mathematically a general kind of such theories, and inferring the inequality bearing his name from characteristic features of such theories by mere logical deduction. In quantum mechanics a corresponding condition can be given, too. These two inequalities are not compatible with one another. Hence it seems to be quite natural to decide between them by experiment. Since all results of experimental investigations from Aspect et al. [3, 4] up to Erven et al. [5] have shown that Bell's inequality is not compatible with the empirical results, local theories nowadays are quite generally refuted. On the other hand all these experiments are supporting the results of quantum mechanics. Hence one could infer that quantum mechanics is correct. But that would be an illegal conclusion, because the agreement with experiment is a necessary, but no su cient condition for the validity of a theory. Hence one would convert a double negation into a positive assertion. In this paper the thesis is claimed and proved that quantum mechanics, indeed, is incomplete, not on the philosophical level, where Bohr and Einstein were quarrelling, but in an immediately physical sense.
This paper discusses the formulation of a quantum mechanical equivalent of the relative time classical theory proposed in Part I. The relativistic wavefunction is derived and a covariant addition theorem is put forward which allows a covariant scattering theory to be established. The free particle eigenfunctions that are given are found not to be plane waves. A covariant partial wave analysis is also given. A means is described of converting wavefunctions that yield probability densities in 4-space to ones that yield the 3-space equivalents. Bound states are considered and covariant analogues of the Coulomb potential, harmonic oscillator potential, inverse cube law of force, square well potential, and two-body fermion interactions are discussed.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world.
Quantum mechanics relates probability of an observable event to the absolute square of the corresponding probability amplitude. It may, therefore, seem that the information about the amplitudes' phases must be irretrievably lost in the experimental data. Yet, there are experiments which report measurements of wave functions, and closely related quantities such as Bohm's velocities and positions of bohmian particles. We invert the question, and ask under which conditions the values of quantum amplitudes can be recovered from observed probability distributions and averages.
The physical meaning of Schrödinger’s wave function remains afoundational interpretive question in quantum mechanics. This paperseparates two questions that are often introduced together: what thewave function describes during propagation, and why experimentaloutcomes obey Born statistics. Rather than taking∣Ψ∣2 as the starting definition of Ψ, we begin from theoperational structure of early electron-scattering and transmissionexperiments surrounding the emergence of wave mechanics and Born’s1926 collision theory. Davisson and Kunsman measured angle-dependent collected current,Ramsauer measured energy-dependent electron transmission throughgases, and Dymond later resolved the energy and angular redistributionof electrons after collisions. In each case, the apparatus returnedphysical signals---charge, current, or flux---from which reproducibleoutcome distributions were reconstructed. From this common structure we propose a deliberately limitedinterpretation: for a single nonrelativistic particle, Ψ may beread first as a complex-amplitude representation of thespatial--directional organization of quantum propagation. Thisproposal is anchored not only in the experimental sequence but also inthe mathematics of the wave function. Its Fourier representationresolves propagation into wave-vector components, while theSchrödinger continuity relation defines a conserved density--currentpair; in polar formΨ=Rexp(iS/ℏ),the associated current is proportional toR2∇S. These facts show that Ψ contains both spatial weighting anddirectional phase structure before any additional ontological claim ismade. We do not infer a material wave medium, deterministic particletrajectories, hidden variables, or an alternative dynamics. Nor do wederive the Born rule. Instead, we isolate a separate mapping problem:how does a structured propagation state, through a specific detectorinteraction, become the observed statistics of localized or collectedparticle events? In scattering, any acceptable account must recover the standard fluxand cross-section relations. The contribution of the present paper istherefore interpretive and programmatic: it identifies a constrainedphysical reading of Ψ and states precisely what remains to beexplained.