### Lecture Description

Leonard Susskind presents an example of rotational symmetry and derives the angular momentum operator as the generator of this symmetry. He then discusses symmetry groups and Lie algebras, and shows how these concepts require that magnetic quantum numbers - i.e. spin - must have whole- or half-integer values.

Professor Susskind presents an example of rotational symmetry and derives the angular momentum operator as the generator of this symmetry. He then presents the concept of degenerate states, and shows that any two symmetries that do not commute imply degeneracy. Symmetries that do not commute can form a symmetry group, and the generators of these symmetries form a Lie algebra.

The angular momentum generators in three dimensions are an example of a symmetry group. Professor Susskind then derives the raising and lowering operators from the angular momentum generators, and shows how they are used to raise or lower the magnetic quantum number of a system between degenerate energy states. Due to reflection symmetry, these states must have whole- or half-integer values for the magnetic quantum number.

Recorded on September 30, 2013.

Topics:

- Rotational symmetry

- Angular momentum

- Commutator

- Degenercy

- Symmetry generators

- Symmetry groups

- Lie algebra

- Raising and lowering operators

### Course Index

- Review of quantum mechanics and introduction to symmetry
- Symmetry groups and degeneracy
- Atomic orbits and harmonic oscillators
- Spin, Pauli Matrices, and Pauli Exclusion Principle
- Fermions: a tale of two minus signs
- Quantum Field Theory: Particle Creation and Annihilation Operators
- Quantum Field Theory: Fermions and Bosons
- Second Quantization
- Quantum Field Hamiltonian
- Fermions and the Dirac equation

### Course Description

This course will explore the various types of quantum systems that occur in nature, from harmonic oscillators to atoms and molecules, photons, and quantum fields. Students will learn what it means for an electron to be a fermion and how that leads to the Pauli exclusion principle. They will also learn what it means for a photon to be a boson and how that allows us to build radios and lasers. The strange phenomenon of quantum tunneling will lead to an understanding of how nuclei emit alpha particles and how the same effect predicts that cosmological space can “boil.” Finally, the course will delve into the world of quantum field theory and the relation between waves and particles.