Operator Formalism

Taku Goto, S. Naka

An operator formalism to the path integral method is developed. In this formalism the operator characterizing the trajectories of the particle is introduced and the usual path integral is expressed in terms of expectation value of a functional of that operator. We also investigate the Wiener-Hermite expansion on the basis of our operator formalism. Feynman's path integral is justified as a sort of analytic continuation of the path integral in the Wiener process.

Norman K. Glendenning

Nikolay Raychev

In this article the concept of phase encoding/decoding is used to analyze and formalize a simple quantum algorithm—the Deutsch’s algorithm. The algorithm is formalized in two different ways through an analysis, based on phase encoding/decoding, carried out by the formalized elementary operators developed by the author of this article. Concrete examples of different possible realizations of the formalized with Raychev’s operators Deutsch’s algorithms are offered.

Willian Wallace Ribeiro Alves da Silva

The main objective of this work is to incorporate angular momentum into the construction of the Position Dependent Translation Operator (PDTO) formalism in the context of quantum mechanics, especially for problems involving rotation. To achieve this goal, we constructed the angular momentum within the conventional formalism of quantum mechanics and also developed the three-dimensional PDTO formalism. In this regard, the construction of angular momentum in the traditional formalism involved the use of operators and equations that describe the behavior of orbital and spin angular momentum. We explored the mathematical properties of these operators to ensure a consistent description of angular momentum within the quantum context. Next, we addressed the construction of the three-dimensional PDTO formalism, taking into account the spatial variations of the properties of the medium in which the particles are located. This involved the generalization of the PDTO equations to the three-dimensional case and the analysis of the physical and mathematical properties of this formalism. Once the angular momentum was constructed within the conventional quantum formalism and the three-dimensional PDTO formalism, the next step was to construct the angular momentum within the PDTO formalism. In this process, we investigated how the mathematical expressions involving angular momentum would be modified and adapted to the new formalism, considering the characteristics of the PDTO and its implications in quantum mechanics. Finally, with the theoretical part established, the ultimate goal is to apply the newly developed formalism to a known physical problem. This will allow us to explore the implications of the PDTO formalism with angular momentum and analyze its consequences for the system under study. Through this application, we can gain new insights, better understand the dynamics of the system, and verify the viability and relevance of the proposed formalism. Thus, it is expected that this work contributes to the advancement of knowledge in quantum mechanics by introducing angular momentum into the PDTO formalism and exploring its theoretical and practical implications. Moreover, this approach may have applications in other areas of physics, such as materials science, quantum optics, and quantum computing, opening new perspectives for future research and scientific and technological developments.