The Unit Commitment (UC) challenge presents a formidable task in mixed integer programming, particularly with large-scale instances. Considering the vast scale of power systems and the substantial resources involved, even minor improvements in solution accuracy and efficiency have the potential to generate significant economic benefits. The quadratic objective function better aligns with the practical realities of the UC problem but solving Mixed Integer Quadratic Programming (MIQP) problems is challenging. This paper introduces a tight quadratic convex reformulation for the UC problem. A novel UC formulation is established after strategically incorporating additional quadratic terms into the objective function. This formulation is tighter than the traditional MIQP UC formulations while maintaining its original structure. We also present its linearized version. A comparative analysis of our formulations against traditional UC formulations confirms their efficiency, showcasing significant computational cost savings and promising prospects for cost reduction.