Professor Susskind elaborates on the abstract mathematics of vector spaces by introducing the concepts of basis vectors, linear combinations of vector states, and matrix algebra as it applies to vector spaces. He then introduces linear operators and bra-ket notation, and presents Hermitian operators as a special class of operators that represent observables. Eigenvectors of Hermitian operators represent orthogonal vector states, and their eigenvalues are the values of the observable. Professor Susskind then applies these concepts to the single spin system that we studied in the last lecture, and introduces the Pauli matrices as the Hermitian operators representing the three spin axis directions. Topics: - Vector spaces and state vectors - Hermitian operators and observables - Eigenvectors and eigenvalues - Normalization and phase factors - Operators for a single spin system - Pauli matrices Recorded on January 23, 2012.

1. Introduction to Quantum Mechanics
2. The Basic Logic of Quantum Mechanics
3. Vector Spaces and Operators
4. Time Evolution of a Quantum System
5. Heisenberg Uncertainty Principle & The Schrödinger Equation
6. Entanglement: Entangled, Singlet, & Triplet States
7. Entanglement and the Nature of Reality
8. Particles Moving in One Dimension and their Operators
9. Fourier Analysis applied to Quantum Mechanics
10. The Uncertainty Principle and Classical Analogs

Quantum theory governs the universe at its most basic level. In the first half of the 20th century physics was turned on its head by the radical discoveries of Max Planck, Albert Einstein, Niels Bohr, Werner Heisenberg, and Erwin Schroedinger. An entire new logical and mathematical foundation—quantum mechanics—eventually replaced classical physics. We will explore the quantum world, including the particle theory of light, the Heisenberg Uncertainty Principle, and the Schrödinger Equation. This course is second-part of a six course sequence given by Prof. Leonard Susskind that explores the theoretical foundations of modern physics - the Theoretical Minimum. Topics in the series include classical mechanics, quantum mechanics, theories of relativity, electromagnetism, cosmology, and black holes.