Abstract: Non-adiabatic level crossings and quantum state transitions are fundamental phenomena occurring across a vast array of physical disciplines. In this talk, I will present a newly derived analytical expression for transition probability, for a two-state quantum system, in the non-adiabatic regime using Density Matrix Formalism (DMF) extending well beyond the traditional Landau-Zener-Stueckelberg-Majorana (LZSM) approximation. While standard LZSM treatment is restricted when considering complex time-dependent mixing or environmental interaction, our DMF framework accommodates both time-dependent dynamics and decoherence effects. Because of its general formulation, this framework applies to any two-state quantum system – offering a powerful analytical tool for particle physics and cosmology (such as active-sterile neutrino mixing, dark matter production and axion-photon mixing), as well as atomic physics, NMR, and quantum information. I will discuss the foundational derivation and illustrate its immediate application to solar neutrino oscillations, while opening the floor to broader prospective applications.