Tobias Osborne: Advanced Quantum Theory (2016)
Many-body quantum mechanics, second quantization, and scattering theory
About This Course
Prof. Tobias Osborne (Leibniz Universität Hannover) presents advanced quantum theory focusing on many-body systems. This 18-lecture course covers second quantization, Fock space construction, quantum field formalism, coherent states, and scattering theory - essential topics bridging single-particle quantum mechanics and quantum field theory.
🎯 Target Audience
Theorists familiar with single-particle QM
📚 Prerequisites
Textbook QM, linear algebra, basic group theory
🔗 Leads To
Quantum field theory, condensed matter theory
Course Outline
Part I: Many-Body QM (1-7)
Identical particles, Fock space, second quantization
7 lecturesPart II: Quantum Fields (8-11)
Field operators, coherent states, variational methods
4 lecturesPart III: Scattering (12-18)
S-matrix, cross sections, Born approximation
7 lecturesAll Lectures
Part I: Many-Body Quantum Mechanics (Lectures 1-7)
Introduction
Course overview, motivation for many-body quantum mechanics, historical context
Video Lecture
Advanced quantum theory, Lecture 1 - Introduction
Introduction to the course and many-body quantum mechanics
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Identical Classical & Quantum Particles
Configuration space of multiple identical classical particles, quantum identical particles
Video Lecture
Advanced quantum theory, Lecture 2 - Identical particles
Study of identical particles in classical and quantum mechanics
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Quantization & Fibre Bundles
Mathematical framework: fibre bundles, connection to gauge theory
Video Lecture
Advanced quantum theory, Lecture 3 - Quantization and fibre bundles
Geometric quantization and bundle structure
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Fock Space Construction (1/2)
Building the infinite-dimensional Hilbert space for variable particle number
Video Lecture
Advanced quantum theory, Lecture 4 - Construction of Fock space (1/2)
First part of Fock space construction
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Fock Space Construction (2/2)
Completion of Fock space construction, symmetrization for bosons and fermions
Video Lecture
Advanced quantum theory, Lecture 5 - Construction of Fock space (2/2)
Second part of Fock space construction
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Fock Space & Dimensions
Properties of Fock space, computing dimensions for various systems
Video Lecture
Advanced quantum theory, Lecture 6 - Fock space and dimensions
Mathematical properties and dimensional analysis
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Second Quantization Procedure
Creation and annihilation operators: a†, a and their commutation relations
Video Lecture
Advanced quantum theory, Lecture 7 - Second quantisation procedure
Introduction to creation and annihilation operators
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Part II: Quantum Fields & Applications (Lectures 8-11)
Second Quantised Hamiltonians
Expressing many-body Hamiltonians using creation/annihilation operators
Video Lecture
Advanced quantum theory, Lecture 8 - Second quantised hamiltonians
Hamiltonian formulation in Fock space
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Quantum Fields & Coherent States
Field operators ψ(x), coherent states |α⟩, and their properties
Video Lecture
Advanced quantum theory, Lecture 9 - Quantum fields and coherent states
Introduction to field operators and coherent states
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Variational Method with Coherent States
Variational principle in Fock space, mean-field approximations
Video Lecture
Advanced quantum theory, Lecture 10 - Variational method over coherent states
Variational methods for many-body systems
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Degenerate Electron Gas
Application to condensed matter: Fermi gas, Pauli exclusion, degenerate matter
Video Lecture
Advanced quantum theory, Lecture 11 - The degenerate electron gas
Many-electron systems and Fermi statistics
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Part III: Scattering Theory (Lectures 12-18)
Introduction to Scattering Theory
Scattering experiments, asymptotic states, in/out states
Video Lecture
Advanced quantum theory, Lecture 12 - Scattering theory
Foundations of quantum scattering theory
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Møller Operators & S-Matrix
Wave operators Ω±, S-matrix definition: S = Ω†₊Ω₋
Video Lecture
Advanced quantum theory, Lecture 13 - Moller operators and the S-matrix
Mathematical formulation of scattering
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
S-Matrix & Cross Sections
Connection to experimental observables: differential cross section dσ/dΩ
Video Lecture
Advanced quantum theory, Lecture 14 - S-matrix and scattering cross section
From S-matrix to measurable cross sections
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
S-Matrix & Green's Operators
Lippmann-Schwinger equation, Green's functions G(E)
Video Lecture
Advanced quantum theory, Lecture 15 - S-matrix and Green's operators
Connection to Green's function formalism
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Green's Operators & Born Approximation
First Born approximation, perturbative scattering amplitudes
Video Lecture
Advanced quantum theory, Lecture 16 - Green's operators and Born approximation
Perturbative methods in scattering theory
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Scattering in Real Space
Position-space representation, scattering amplitudes f(θ,φ)
Video Lecture
Advanced quantum theory, Lecture 17 - Scattering process in real space
Real-space formulation of scattering
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Symmetries & Wigner's Theorem
Symmetry transformations in quantum mechanics, proof of Wigner's theorem
Video Lecture
Advanced quantum theory, Lecture 18 - Symmetries and Wigner's theorem
Fundamental theorem on symmetry representations
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Course Connections
Prerequisite to QFT: Second quantization is the foundation for quantum field theory. After this course, proceed to Tobias Osborne's QFT 2016 for relativistic fields.
Condensed Matter Physics: Fock space formalism and second quantization are essential for solid state physics, superconductivity, and quantum many-body theory.
Scattering Theory: Connects to experimental particle physics, nuclear physics, and provides foundation for LSZ reduction formula in QFT.
Mathematical Physics: Fibre bundles (Lecture 3) connect to gauge theory and geometric quantization, important for advanced theoretical physics.
Key Concepts Covered
Mathematical Structures
- • Fock space: ℱ = ⊕ₙ ℋ⊗ⁿ
- • Creation operators: a†|n⟩ = √(n+1)|n+1⟩
- • Annihilation operators: a|n⟩ = √n|n-1⟩
- • [a,a†] = 1 (bosons) or {a,a†} = 1 (fermions)
- • Field operators: ψ(x) = ∑ₖ aₖφₖ(x)
Physical Applications
- • Many-electron systems (Pauli exclusion)
- • Bose-Einstein condensation
- • Scattering cross sections
- • S-matrix and transition amplitudes
- • Born approximation for weak potentials