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.
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.
The present work aimed to discuss quantum mechanics in phase space, propose a modification of the Wigner function through the position-dependent translation operator (PDTO) and apply the proposed theory to systems of physical interest. Thus, using a one-dimensional non-Euclidean metric, we construct a modified Wigner function whose form can be expressed in terms of the position-dependent translation operator for the case of finite translations. Through this metric, we define a point transformation that acts as a parameterization that preserves the Euclidean form of the equations. Thus, it was possible to find a modified expression for the Weyl-Wigner quantization map, whose objective is to map functions in phase space into operators in Hilbert space. We show that, when using this modified map to quantize the classical momentum, we obtain the same result as the modified momentum operator found via the position-dependent translation operator. Furthermore, through the inverse operation of the parameterized Weyl-Wigner map, it was possible to define a modified form of the Wigner function, whose mathematical form from the Euclidean case was preserved. We apply the proposed Wigner function to the particle trapped in the potential well and to a quantum oscillator-type system, with the aim of evaluating the influence of the metric on the system dynamics. We plotted the graph of the modified Wigner function for these two systems, where it was possible to verify the appearance of deformations due to the influence of the metric used. Thus, we concluded that the dynamics of the system undergoes a change when the structure of the space is deformed.
In this chapter, we develop the background needed to move our quantum mechanical formalism fully into three dimensions. We discuss angular momentum, which is the motional complement to spin angular momentum, that we developed earlier. There are some critical differences between orbital and spin angular momentum, and we need to carefully discuss how they relate and how they differ. Why is angular momentum so important? Because in problems where we have central forces (forces that depend only on the distance between two particles and act along the line connecting the two particles), the angular momentum is conserved. So, it becomes an integral part of how we solve the problem. The most important problem we have to solve, the hydrogen atom, is a central force problem; hence, we need this work before we can move forward to some really interesting physics that will come in Chap. 11 . We cover the angular momentum operators, eigenvalues, and eigenstates, their matrix representations, how to combine angular momentum, and spherical harmonics. We revisit the EPR experiment using our new formalism. We also discuss some unique representations of angular momentum by mapping to two-dimensional harmonic oscillators, as developed by Schwinger.
This chapter begins by developing the relation between the quantum theory and the linear algebra. It discusses the most fundamental postulates of the theory and provides a phenomenological development of the Schrödinger wave equation (SWE) although the results might be gleaned from the classical Lagrangian and Hamiltonian with the Poisson bracket relations. Afterward some simple examples for the infinitely and finitely deep wells help clarify the basic postulates. The simple harmonic oscillator has perhaps the most sweeping implications and applications for the quantum theory. For this reason, we present the most applicable formalism for the Harmonic oscillator that uses the operator approach rather than the classical method of partial differential equations. The angular momentum and spin are presented for the study of the atom and for applications for nanodevices, and quantum computing. Next the representation formalism clarifies the distinction between the dynamical operators and the wave functions and their interrelations. The chapter discusses both timeindependent and time-dependent perturbation theories. The density operator combines the quantum theory with the classical theory for finding the behavior of a system. The density operator represents one of the basic concepts in the quantum theory. The remainder of the chapter introduces the more advanced material meant to introduce the quantum field theory starting with the second quantization and the propagator.
In this work we analyze some aspects related to the introduction, in the non-relativistic quantum mechanics,of a translation operator whose action is a function of the point of application. From the introduction of this operator, we derive a modified canonical commutation relation and a modified representation for the momentum operator and we show that such changes require a restructuring of the Hilbert space. We also see that the “new” Hilbert space can be associated with the case of a particle with position dependent mass and also that of a particle moving in a curved space. Still, we compare our formalism with the formalism of von Roos, we analyze the classical limit of this formalism,we solve the problem of a particle with position-dependent mass in two dimensions and we end up analyzing some quantities of information theory for the case of a one-dimensional harmonic particle.
The theory of angular momentum performance a significant position in the classical and quantum mechanical study of physical properties, such as studies into nuclear, atomic, and molecular processes, as well as other quantum problems, including spherical symmetry. In this analysis, angular momentum operators are described in multiple ways, based on the angular momentum operator's commutator, matrix, and geometric representation, The eigenvalue and eigenvector were also known for operatorsJ ̂_±,J ⃑ ̂^2, J ̂_x,J ̂_y and J ̂_zwithin the |j,┤ ├ m⟩ basis. Furthermore, in quantum mechanics, angular momentum is called quantized variable, meaning that it comes in discrete quantities. In contrast to the macroscopic system case where a continuous variable is angular momentum.
Abstract This paper is concerned chiefly with the introduction of a co-ordinate operator in the quantum mechanics of the electron. It differs from other work on the same subject in the derivation, of the form of the operator by an analogy with the quantum expression for the angular momentum of the electron. Expressions for velocity and momentum operators are derived, and the relation between them is formally that of the classical theory in the absence of a field of force. By the introduction of reciprocal relations for the co-ordinate and momentum operators a condition is obtained which is applicable when the finite extension of the particle cannot be ignored. This is similar to the condition introduced by Yukawa in the theory of non-local fields. In the course of the development of the work, an explanation is offered of the significance of the fifth co-ordinate introduced in Kaluza’s theory of the gravitational and electromagnetic field and in Klein’s adaptation of it. Further, a new aspect of the question of the quantization of space and time is revealed by the use of co-ordinate operators (Xk), for it appears that the interest lies, not in a property of space-time, but in the quantization of localization of a particle. It should, however, be emphasized that the co-ordinates (xk) are retained with their usual meaning, but it is suggested that they are insufficient as a basis of a mechanical description of the electron. It is by operations upon (Xk) and not upon (xk) that mechanical quantities are obtained. The momentum (uk), like (xk), keeps its usual meaning, but it also is part of an operator (Mk) of wider significance. This is analogous to the case of the angular momentum of the electron where Dirac’s introduction of ‘spin’ shows that the orbital angular momentum is a part of a complete angular momentum operator. The application has been limited to the electron, because the derivation of the form of the operator (Xk) rests upon this analogy. The intention has not been to develop a calculus in which ordinary geometrical co-ordinates are replaced by operators. Finally, in proceeding to a force operator an additional term arises proportional to the derivative of the acceleration with respect to time. This term, which is usually ascribed to the reaction of the field of the electron, occurs here without reference to the self field, and suggests that, in addition to charge and current, a non-Maxwellian quantity must be considered as interacting with the external field.