Professor Susskind opens the lecture by presenting the four fundamental principles of quantum mechanics that he touched on briefly in the last lecture. He then discusses the evolution in time of a quantum system, and describes how the classical concept of reversibility relates to the quantum mechanical principle of conservation of information, which is actually the conservation of distinctions or distinguishability of states. The evolution in time of a quantum system is represented by unitary operators which preserve distinctions and overlap. Professor Susskind then derives the time-dependent Schrödinger equation, and describes how to calculate the expected value of an observable, and how it changes with time. This discussion introduces the commutator operator. Professor Susskind closes the lecture by showing the connection between the quantum mechanical commutator and the Poisson bracket formulation of classical physics, thus showing how the time evolution of the expected value of an observable is closely related to classical equations of motion. Topics: - Four fundamental principles of quantum mechanics - Unitarity and unitary evolution of a system - Reversibility, conservation of information, preservations of distinctions, and conservation of overlap of states - Derivation of the time-dependent Schrödinger equation - Time evolution of expectation value and equivalence to classical equations of motion - Parallel between quantum mechanical commutator and classical Poisson bracket Recorded on January 30, 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.