This chapter provides a brief introduction to Quantum Mechanics (QM) for mathematicians, focusing on algebraic aspects and the representation of QM on Hilbert spaces. It begins by highlighting the broad applicability of QM, emphasizing that quantum effects are significant at small scales governed by the Planck constant $\hbar$. The historical development of QM is discussed, including the work of von Neumann and Mackey, and references to modern texts for further study. The chapter then delves into the mathematical framework of QM, defining physical systems and their observables using Segal's approach. It contrasts classical mechanics, where observables form commutative algebras, with quantum mechanics, where observables form non-commutative algebras. The Heisenberg uncertainty principle is introduced as a key feature of quantum mechanics, and the concept of states is explored, including pure and mixed states. The chapter also discusses the quantization process, exemplified by the Weyl algebra, and the Schrödinger representation, which provides a concrete realization of the abstract algebra of observables in a Hilbert space. The dynamics of quantum systems is described using algebraic dynamical systems, and the Heisenberg and Schrödinger equations are derived to describe the time evolution of observables and states, respectively. Finally, the chapter touches on the quantum particle in a potential, discussing the well-posedness of the Schrödinger equation for potentials in the Kato class, and the application of these results to problems in atomic and nuclear physics, such as the hydrogen atom.This chapter provides a brief introduction to Quantum Mechanics (QM) for mathematicians, focusing on algebraic aspects and the representation of QM on Hilbert spaces. It begins by highlighting the broad applicability of QM, emphasizing that quantum effects are significant at small scales governed by the Planck constant $\hbar$. The historical development of QM is discussed, including the work of von Neumann and Mackey, and references to modern texts for further study. The chapter then delves into the mathematical framework of QM, defining physical systems and their observables using Segal's approach. It contrasts classical mechanics, where observables form commutative algebras, with quantum mechanics, where observables form non-commutative algebras. The Heisenberg uncertainty principle is introduced as a key feature of quantum mechanics, and the concept of states is explored, including pure and mixed states. The chapter also discusses the quantization process, exemplified by the Weyl algebra, and the Schrödinger representation, which provides a concrete realization of the abstract algebra of observables in a Hilbert space. The dynamics of quantum systems is described using algebraic dynamical systems, and the Heisenberg and Schrödinger equations are derived to describe the time evolution of observables and states, respectively. Finally, the chapter touches on the quantum particle in a potential, discussing the well-posedness of the Schrödinger equation for potentials in the Kato class, and the application of these results to problems in atomic and nuclear physics, such as the hydrogen atom.