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The extension of Standard Model of the Fundamental Particles which considerthe Lorentz Symmetry Violation governed by a background tensor field presentedby Colladay and Kostelecky consider the hypothesis of having a privilegeddirection in the space-time. Here we investigated a extension of theSchr\"odinger Equation inspired on that model taking into account a backgroundscalar field. We analyze the behavior of this modified Schr\"odinger equationon a particle trapped in a well and quantum harmonic oscillator.

Effects of a background scalar field induced by the Lorentz symmetry violation on Non Relativistic Quantum Mechanics