By means of the usual definition of inner product and of the Gauss condition on the sum of squares of three integers in number theory, it can be seen that there exist specific directions in continuous space in which a classical point particle moving in a three-dimensional lattice cannot propagate. When representing all directions for which propagation is possible as points on a unitary sphere, the forbidden directions appear as vacancies on this sphere. By means of a stereographic projection of the allowed direction, it is argued that propagation is not allowed for specific sets of points on the stereographic plane. The present work can be considered as an interdisciplinary lecture for advanced high-school students or to first-year college students.
Straight-line motion of classical point particles in a three-dimensional lattice
DE LUCA, Roberto
2016-01-01
Abstract
By means of the usual definition of inner product and of the Gauss condition on the sum of squares of three integers in number theory, it can be seen that there exist specific directions in continuous space in which a classical point particle moving in a three-dimensional lattice cannot propagate. When representing all directions for which propagation is possible as points on a unitary sphere, the forbidden directions appear as vacancies on this sphere. By means of a stereographic projection of the allowed direction, it is argued that propagation is not allowed for specific sets of points on the stereographic plane. The present work can be considered as an interdisciplinary lecture for advanced high-school students or to first-year college students.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.