We consider a modified non-relativistic quantum mechanics where the position and momentum operators satisfy a non-standard commutation relation of the form $[X_i, P_j] = i\hbar\ {(1+\beta P^2) + \beta' P_iP_j\}$. Such a theory incorporates an absolute minimal length, UV/IR mixing and non-commutative position space. The possible representations in terms of differential operators are analyzed and their equivalence to first order is established.
Simple quantum systems, namely the harmonic oscillator, the Coulomb potential and the gravitational well are studied in one of these representations, the pseudo-position one, and results are compared to previously published results. The Coulomb potential is also analyzed by an alternative analytical/numerical method. A constraint of $\sim 3$GeV on the scale of the parameters $\beta, \beta'$ is obtained from precision experimental data on the atomic hydrogen energy levels.
