The conventional quantum mechanical problem of a particle in a one-dimensional box can be made more interesting chemically without introducing any more difficult mathematics by employing a potential diagram rather than the usual infinite well.
Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided.
We consider the -deformed formalism constructed with some elements of Fibonacci calculus to study the 1D infinite potential box problem where 1-D Schrdinger equation is reframed with well-known modified Fibonacci difference operator and has been solved using q-series solution method. In order to do it, at first, all the selective but essential basics are displayed one by one and -deformed trigonometric functions have been plotted to review and understand the deformed quantum mechanical framework. Hence, on the basis of two parameter deformed algebra, wave functions associated with the particle confined inside the infinite potential box has been obtained. The normalized wave functions and energy expressions have also been obtained for . It is found that all the expressions can be reduced to their conventional form within the limit .