Abstract: Quantum mechanics, a cornerstone of modern physics, has undergone significant theoretical development since its emergence in the early 20th century. This paper classifies quantum mechanical frameworks into three categories—core formalisms, physical interpretations, and extended tools—and examines 14 key frameworks, spanning from matrix mechanics (1925) to Bayesian quantum mechanics (21st century). Each framework is analyzed in historical context, elucidating its theoretical foundations, mathematical formulations, and practical applications,with analogies to clarify complex concepts. The study traces the evolution of quantum mechanics from singleparticle to manybody systems and from concrete computations to abstract algebraic structures, highlighting its profound influence on mathematics, philosophy, and quantum information science.

Key words: quantum Mechanics, wave Function, interpretation