The harmonic oscillator in Snyder space is investigated in its classical and quantum versions. The classical trajectory is obtained and the semiclassical quantization from the phase space trajectories is discussed. An effective cut-off to high frequencies is found. The quantum version is developed and an equivalent usual harmonic oscillator is obtained through an effective mass and an effective frequency introduced in the model. This modified parameters give us a modified energy spectrum also.

In this paper the Kepler problem in the non-commutative Snyder scenario was studied. The deformations were characterized in the Poisson bracket algebra under a mimic procedure from quantum standard formulations by taking into account a general recipe to build the noncommutative phase space coordinates (in the sense of Poisson brackets). An expression for the deformed potential was obtained, and then the consequences in the precession of the orbit of Mercury were calculated. The result could be used for finding an estimated value for the non-commutative deformation parameter.

The lack of experimental results of theories concerning the ultimate space/space–time structure, leads one to think about some simple experiments that can help demonstrate the main prediction: Space–time is discrete. In this paper, an implementation of Snyder operators is applied to simple geometries, to demonstrate that the spectrum of the position operator, in standard quantum mechanics (QM), is discrete when a parameter measuring the non-commutativity of quantum operators is introduced. The geometries are specially suitable for experiments in search of the behaviour of ultracold neutrons falling in the Earth’s gravitational potential.