Year
Publisher
Journal
1
Institution
Institution Country
Publication Type
Field Of Study
Topics
Open Access
Language
Filter 1
Year
Publisher
Journal
1
Institution
Institution Country
Publication Type
Field Of Study
Topics
Open Access
Language
Filter 1
Export
Sort by: Relevance
Quasiclassical approximation does not have to be applied to all degrees of freedom: the solution for the S-states of singular attractive potentials

For the spherically symmetric potentials V(r), the quasiclassical approximation (QA) usually is applied as follows. For the φ-motion, the QA is not applied: instead, the exact solution is used. The QA is then applied to both the θ-motion and the r-motion. The resulting QA wave function for the θ-motion is valid for some, but not all values of θ. Besides, the resulting QA wave function for the θ-motion is valid only with the substitution of L(L+1) by (L+1/2)2, where L is the angular momentum quantum number. Then the QA is applied to the r-motion, but again with the substitution of L(L+1) by (L+1/2)2 in the centrifugal energy term. As a result of the above standard procedure, the obtained QA wave functions are valid only for (L + 1/2) greater or of the order of N1/2, where N is the principal quantum number. Besides, for potentials having the pole of the order 3 or higher at r = 0 or for relatively large r, it would be preferable to have L(L + 1) without the substitution by (L+1/2)2. This is yet another restriction of the QA validity in this procedure. In the first part of the present paper, we provide a better alternative. This alternative can also be applied to some potentials whose geometrical symmetry is not spherical. In the second part of the present paper, we focus on the attractive singular spherically-symmetric potentials V(r) =-an/rn, where an > 0 and the integer n > 2. They are encountered in many physical systems. There exists a paradigm that “singular attractive potentials do not lead to physically reasonable results” because eigenfunctions depend on the chosen cutoff at small r. In the present paper we disprove this paradigm. We demonstrate that while the unnormalized eigenfunctions of singular attractive potentials do depend on the cutoff at small r, there is no such problem for the corresponding normalized eigenfunctions. Namely, the normalized eigenfunctions, corresponding to any two different sufficiently small cutoffs (all other parameters being the same), are equal to each other or differ just by the sign. The sign difference does not affect the probability density: the latter is invariant to the variations of the cutoff. Thus, we prove that singular attractive potentials actually do lead to physically reasonable results. In doing so, we also demonstrated that a divergent integral can lead to physically meaningful results independent of the cutoff. This is a counterintuitive result of the importance from the fundamental point of view both for physics and mathematics.

Read full abstract
Open Access
Towards a probabilistic foundation of relativistic quantum theory: the one-body Born rule in curved spacetime

In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.

Read full abstract
Open Access