Textbook in PDF format The modern version of quantum mechanics was formulated in 1932 by John von Neumann in his famous book Mathematical Foundations of Quantum Mechanics, and it unifies Schrödingers wave theory with the matrix mechanics of Heisenberg, Born, and Jordan. The theory is framed in terms of linear vector spaces, so we also have to remind ourselves of the relevant mathematics. The entire structure of quantum mechanics (including its relativistic extension) can be formulated in terms of states and operations in Hilbert space. We need rules that map the physical quantities such as states, observables, and measurements to the mathematical structure of vector spaces, vectors and operators. There are several ways in which this can be done. Linear Vector Spaces and Hilbert Space The Postulates of Quantum Mechanics Schrödinger and Heisenberg Pictures Mixed States and the Density Operator Composite Systems and Entanglement Evolution of Open Quantum Systems Orbital Angular Momentum and Spin Identical Particles Many-Body Problems in Quantum Mechanics Pauli Spin Matrices Noncommuting Operators and Uncertainty Multiple Particle States The Schrödinger Equation Atomic Orbitals The Periodic Table of the Elements Matter