**Copyright Information:**All rights reserved to Prof. Leonard Susskind, Stanford University.

### Lecture Description

Professor Susskind begins the lecture with a review of the problem of a single spin in a magnetic field. He re-emphasizes that observables corresponding to the Pauli sigma matrices do not commute, which implies that they obey the uncertainty relationship, and reviews the principles by which the spin in a magnetic field will radiate a photon and transition to the lowest possible energy state. Professor Susskind then moves on to discuss the effect of measurement on a quantum system and the concept of wave function collapse. In general, the measuring apparatus becomes part of the quantum system and the space of states for the combines system is the tensor product of the states of the individual system components. This is the concept of entanglement. Professor Susskind demonstrates the simplest example of entanglement of a two spin system. He distinguishes the unentangled product states from the more general entangled states, and gives examples or operators and expectation values for each. The singlet and triplet states are introduced. Professor Susskind concludes the lecture by summarizing the essence of entanglement in the principle that, although a single spin quantum mechanical system can be simulated with a classical computer, a two spin system cannot be simulated by two classical computers unless they are connected together. Topics: - Wave function collapse - Tensor products - Product states - Entanglement - Observables for entangled states - Expectation values of entangled states - Singlet and triplet states Recorded on February 13, 2012.

### Course Index

- Introduction to Quantum Mechanics
- The Basic Logic of Quantum Mechanics
- Vector Spaces and Operators
- Time Evolution of a Quantum System
- Heisenberg Uncertainty Principle & The Schrödinger Equation
- Entanglement: Entangled, Singlet, & Triplet States
- Entanglement and the Nature of Reality
- Particles Moving in One Dimension and their Operators
- Fourier Analysis applied to Quantum Mechanics
- The Uncertainty Principle and Classical Analogs

### Course Description

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.