This text provides an introduction to Quantum Mechanics (QM) for mathematicians, focusing on its algebraic structure and mathematical foundations. It discusses the historical development of QM, its mathematical framework, and the role of Hilbert spaces. The text explains the basic principles of QM, such as the representation of physical observables as self-adjoint operators, the definition of quantum states as functionals on the space of observables, and the time evolution of quantum systems through self-adjoint operators. It also contrasts QM with Classical Mechanics, highlighting the Heisenberg uncertainty principle and the non-commutative nature of quantum observables. The text introduces the concept of C*-algebras as a mathematical framework for QM, and discusses the GNS theorem, which provides a representation of QM in terms of Hilbert spaces. It also covers the Weyl algebra, which is used to describe the quantum particle, and the Schrödinger representation, which is a specific way of representing the Weyl algebra. The text concludes with a discussion of algebraic dynamics, the Heisenberg and Schrödinger equations, and the quantum Hamiltonian. The text references several key mathematical texts on QM, including those by von Neumann, Mackey, Strocchi, and Isham.This text provides an introduction to Quantum Mechanics (QM) for mathematicians, focusing on its algebraic structure and mathematical foundations. It discusses the historical development of QM, its mathematical framework, and the role of Hilbert spaces. The text explains the basic principles of QM, such as the representation of physical observables as self-adjoint operators, the definition of quantum states as functionals on the space of observables, and the time evolution of quantum systems through self-adjoint operators. It also contrasts QM with Classical Mechanics, highlighting the Heisenberg uncertainty principle and the non-commutative nature of quantum observables. The text introduces the concept of C*-algebras as a mathematical framework for QM, and discusses the GNS theorem, which provides a representation of QM in terms of Hilbert spaces. It also covers the Weyl algebra, which is used to describe the quantum particle, and the Schrödinger representation, which is a specific way of representing the Weyl algebra. The text concludes with a discussion of algebraic dynamics, the Heisenberg and Schrödinger equations, and the quantum Hamiltonian. The text references several key mathematical texts on QM, including those by von Neumann, Mackey, Strocchi, and Isham.