A momentum operator for non-equidistant and non-periodic partitions

Ponente(s): José Armando Martínez Pérez, Dr. Gabino Torres Vega (Investigador Cinvestav, Departamento de Física)

The properties of operators with continuous spectra are well developed. However, only a few things are known about discrete operators. We introduce a momentum-like operator for a non-periodic and non-equidistant partition as spectrum, in the continuous case it is the derivative multiplied by i, the imaginary number. The equidistant and periodic case, on the unit circle, has been studied before and it was shown that the commutator between the coordinate and momentum operators approach the identity in the appropriate limit. Here, we are interested in a discrete momentum operator for non-periodic and non-equidistant meshes. The properties of our operator are similar to the continuous operator, without the need of any limit. Among other things, we found the eigenvectors and its properties, and a discrete version of the uncertainty principle is developed. Our results still allow us to study discrete quantum systems very similar to the continuous ones