Leonard Susskind Lectures

The course begins with a brief review of quantum mechanics and the material presented in the core Theoretical Minimum course on the subject. The concepts covered include vector spaces and states of a system, operators and observables, eigenfunctions and eigenvalues, position and momentum operators, time evolution of a quantum system, unitary operators, the Hamiltonian, and the time-dependent and independent Schrodinger equations. After the review, Professor Susskind introduces the concept of symmetry. Symmetry transformation operators commute with the Hamiltonian. Continuous symmetry transformations are composed from the identity operator and a generator function. These generator functions are Hermitian operators that represent conserved quantities. The lecture closes with the example of translational symmetry. The generator function for translational symmetry is the momentum operator divided by ħ. Topics: Vector space Observables Hermitian operators Eigenvectors and eigenvalues Position and momentum operators Time evolution Unitarity and unitary operators The Hamiltonian Time-dependent and independent Schrödinger equations Symmetry Conserved quantities Generator functions

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. Topics: Rotational symmetry Angular momentum Commutator Degenercy Symmetry generators Symmetry groups Lie algebra Raising and lowering operators

Professor Susskind uses the quantum mechanics of angular momentum derived in the last lecture to develop the Hamiltonian for the central force coulomb potential which describes an atom. The solution of the Schrödinger equation for this system leads to the energy levels for atomic orbits. He then derives the equations for a quantum harmonic oscillator, and demonstrates that the ground state of a harmonic oscillator cannot be at zero energy due to the Heisenberg uncertainty principle. Topics: Angular momentum multiplets Coulomb potential Central force problem Atomic orbit Harmonic oscillator Heisenberg uncertainty principle

Professor Susskind builds on the discussion of quantum harmonic oscillators from the last lecture to derive the higher order energy states and wave functions. He then moves on to discuss spin states of particles, and introduces the Pauli matrices, which account for the interaction of a particle's spin with an external magnetic field. By examining the energy levels of electrons in an atom, Pauli and others realized that only two electrons can be in any given state. This led both to the the exclusion principle, as well as the need for another state variable - spin - which allows two electrons in each energy level. Topics: Spin Pauli matrices Pauli exclusion principle

Professor Susskind presents the quantum mechanics of multi-particle systems, and demonstrates that fermions and bosons are distinguished by the two possible solutions to the wave function equation when two particles are swapped. When two particles are swapped, the boson wave function equation has a phase factor of +1 whereas the fermion equations has a phase factor of -1. For fermions, this results in a wave function with zero probability for two particles to be in the same state, thus demonstrating the exclusion principle. On the other hand, bosons prefer to be in the same state. This is what makes a photon (boson) laser possible, but an electron (fermion) laser impossible. The spin variable is required to allow two electrons to occupy the same state in an atom. Electrons are fermions which have half-integer spins. This implies that a rotation of the angular momentum by 2π will result in a phase change by -1. This implies that the identity operation for fermions is not a rotation by 2π, but rather a rotation by 4π, and that a rotation by 2π can be offset or canceled by a swap of two particles. This is the tale of 2 minus signs. Topics: Bosons Fermions Spin statistics Permutation groups Solitons

Professor Susskind introduces quantum field theory. Excepting gravity, quantum field theory is our most complete description of the universe. Each quantum field corresponds to a specific particle type, and is represented by a state vector consisting of the number of particles in each possible energy state. These numbers are called occupation numbers. This representation uses the same quantum mathematics as a state vector for multiple harmonic oscillators with the basis vectors being the energy state of each oscillator. Therefore the quantum field mathematics follow those introduced for harmonic oscillators in previous lectures. However, in the case of quantum field theory, the raising and lowering operators become operators which create and destroy particles in a given energy state. Topics: Quantum field theory Occupation numbers Creation and anhilation operators

Professor Susskind continues with the presentation of quantum field theory. He reviews the derivation of the creation and annihilation operators, and then develops the formulas for the energy of a multi-particle system. This derivation demonstrates the correspondence between classical and quantum field theory for many particle systems. Topics: Classical field theory

Professor Susskind answers a question about neutrino mixing and relates the oscillating quantum states of a neutrino to a precessing electron spin in a magnetic field. He then discusses a recent article about whether an electron is a sphere. After these topics, Professor Susskind continues the previous discussion about second quantization, and demonstrates that the position and momentum creation and annihilation operators are Fourier conjugates of each other. Topics: Neutrino mixing Second quantization Fourier conjugates

Professor Susskind presents the Hamiltonian for a quantum field, and demonstrates how these Hamiltonians describe particle interactions such as decay and scattering. He then introduces the field theory for fermions by deriving the Dirac equation. The theory behind the Dirac equations was the first theory to account fully for special relativity in the context of quantum mechanics. This relativistic Schrödinger equation implies the existence of antimatter. Topics: Hamiltonian Dirac equation Klein-Gordon equation Antimatter

Professor Susskind closes the course with the presentation of the quantum field theory for spin-1/2 fermions. This theory is based on the Dirac equation, which, when Dirac developed it in 1928, was the first thory to account fully for special relativity in the context of quantum mechanics. This theory explains spin as a consequaence of of the union of quantum mechanics and relativity, and also led to the theory of antimatter and ultimate discovery of the first antimatter particle - the positron. Professor Susskind begins the presentation by reviewing the Dirac equation for an electron in one dimension, and then generalizes this to derive the therory for three dimensions. This led Dirac to develop his 4x4 gamma matrices. In Dirac's theory, the mass of fermions originates from the cross term between the two chiralities in the Dirac equation. Topics: Fermion Dirac equation Pauli matrices Chirality Dirac sea Zitterbewegung Positron