Modified Heisenberg Commutation Relations and the Infinite-Square Well Potential: Some Simple Consequences

We explore some consequences of modifying the usual Heisenberg commutation relations of two simple systems: first, the one-dimensional quantum system given by the infinite square-well potential, and second, the case of a gas of N non-interacting particles in a box of volume V, which permit obtaining analytical solutions. We analyse two possible cases of modified Heisenberg commutation relations: one with a linear and non-linear dependence on the position and another with a linear and quadratic dependence on the momentum. We determine the eigenfunctions, probability densities, and energy eigenvalues for the one-dimensional square well for both deformation cases. For linear and non-linear x deformation dependence, the wave functions and energy levels change substantially when the weight factor associated with the modification term increases. Here, the energy levels are rescaled homogeneously. Instead, for linear and quadratic momentum p deformation dependence, the changes in the energy spectrum depend on the energy level. However, the probability densities are the same as those without any modification. For the non-interacting gas, the position deformation implies that the ideal gas state equation is modified, acquiring the form of a virial expansion in the volume, whereas the internal energy is unchanged. Instead, the ideal gas state equation remains unchanged at the lowest order in β for the momentum modification case. However, the temperature modifies the internal energy at the lowest order in β. Thus, this study indicates that gravity could generate forces on particles by modifying the Heisenberg commutation relations. Therefore, gravitation could be the cause of the other three forces of nature.