Circular quantum dots within the position dependent translation operator formalism - Pimentel, Gabriel Araújo de Oliveira (2024)
Quantum mechanics allows the study of the behavior of the interaction of radiation with matter on a subatomic scale. The second postulate of quantum mechanics states that every measurable physical quantity is described by operators acting on the space of state vectors. In this sense, the definition of a translation operator allows quantum states to be moved to different points in space and makes it possible to study quantum dynamics phenomena. In 2011, Costa Filho et al. presented a new mathematical formalism for the translation operator, making it positiondependent, with the aim of studying particle systems with position-dependent mass and, thus, the effects of space metrics on quantum theory. Here, we present the formalism of the Position Dependent Translation Operator in general coordinates and use it to solve the problem of an isotropic harmonic oscillator in polar coordinates, i.e., to study the effects of two different metrics on the confinement of electrons in the circular quantum dot. To do this, we obtain the energies of the confined states and their respective wave functions depending on a metric factor. Results for different metric deformation values are discussed and compared with the case of the usual quantum harmonic oscillator.