💫Intro to Quantum Mechanics II Unit 10 – Multi-Particle Systems & 2nd Quantization

Key Concepts

• Quantum many-body systems consist of multiple interacting particles described by quantum mechanics
• Identical particles are indistinguishable and exhibit symmetry under exchange (bosons or fermions)
• Fock space is a Hilbert space representation for variable number of particles using occupation numbers
• Creation and annihilation operators add or remove particles from specific quantum states
• Second quantization reformulates quantum mechanics in terms of field operators acting on Fock space
  • Enables efficient description of many-body systems and their interactions
• Quantum field theory extends these concepts to relativistic systems and fundamental particles
• Problem-solving strategies involve identifying symmetries, using commutation relations, and applying perturbation theory

From Classical to Quantum Many-Body Systems

• Classical many-body systems involve Newtonian mechanics and interactions via forces
• Quantum many-body systems require quantum mechanics to accurately describe particle behavior
  • Particles exhibit wave-particle duality and are governed by the Schrödinger equation
• Quantum effects become significant at small scales (atomic and subatomic) and low temperatures
• Many-body quantum systems exhibit emergent phenomena not present in classical systems (superfluidity, superconductivity)
• Interactions between particles are modeled using potential energy terms in the Hamiltonian
• Statistical mechanics connects microscopic quantum behavior to macroscopic thermodynamic properties
• Quantum entanglement plays a crucial role in the collective behavior of many-body systems

Identical Particles and Symmetry

• Identical particles are indistinguishable and share the same intrinsic properties (mass, charge, spin)
• Exchanging identical particles results in a symmetric or antisymmetric wavefunction
  • Bosons have symmetric wavefunctions and integer spin (photons, gluons, Higgs boson)
  • Fermions have antisymmetric wavefunctions and half-integer spin (electrons, quarks, neutrinos)
• The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state
• Bose-Einstein and Fermi-Dirac statistics describe the distribution of bosons and fermions in thermal equilibrium
• Symmetrization and antisymmetrization operators project wavefunctions onto the appropriate symmetry subspace
• Permutation operators exchange particle labels and are used to classify symmetry types

Fock Space and Occupation Number Representation

• Fock space is a Hilbert space that accommodates a variable number of particles
  • Each basis state represents a specific configuration of particles occupying different quantum states
• Occupation numbers $n_i$ indicate the number of particles in each single-particle state $|i\rangle$
• A Fock state $|n_1, n_2, \ldots\rangle$ describes a system with $n_1$ particles in state $|1\rangle$, $n_2$ in $|2\rangle$, etc.
• The vacuum state $|0\rangle$ represents a system with no particles
• Fock states form a complete orthonormal basis for the many-body Hilbert space
• Operators in Fock space act on occupation numbers rather than individual particle coordinates
• The total number operator $\hat{N} = \sum_i \hat{n}_i$ counts the total number of particles in the system

Creation and Annihilation Operators

• Creation operators $\hat{a}^\dagger_i$ add a particle to the single-particle state $|i\rangle$
  • Acting on a Fock state: $\hat{a}^\dagger_i |n_1, \ldots, n_i, \ldots\rangle = \sqrt{n_i + 1} |n_1, \ldots, n_i+1, \ldots\rangle$
• Annihilation operators $\hat{a}_i$ remove a particle from the single-particle state $|i\rangle$
  • Acting on a Fock state: $\hat{a}_i |n_1, \ldots, n_i, \ldots\rangle = \sqrt{n_i} |n_1, \ldots, n_i-1, \ldots\rangle$
• Creation and annihilation operators satisfy commutation relations
  • Bosons: $[\hat{a}_i, \hat{a}^\dagger_j] = \delta_{ij}$, $[\hat{a}_i, \hat{a}_j] = [\hat{a}^\dagger_i, \hat{a}^\dagger_j] = 0$
  • Fermions: $\{\hat{a}_i, \hat{a}^\dagger_j\} = \delta_{ij}$, $\{\hat{a}_i, \hat{a}_j\} = \{\hat{a}^\dagger_i, \hat{a}^\dagger_j\} = 0$
• The number operator for a single-particle state is $\hat{n}_i = \hat{a}^\dagger_i \hat{a}_i$
• Creation and annihilation operators enable compact expressions for many-body operators and interactions

Second Quantization Formalism

• Second quantization reformulates quantum mechanics in terms of field operators acting on Fock space
• Field operators $\hat{\psi}(\mathbf{r})$ and $\hat{\psi}^\dagger(\mathbf{r})$ create or annihilate particles at position $\mathbf{r}$
  • Expanded in terms of single-particle wavefunctions: $\hat{\psi}(\mathbf{r}) = \sum_i \phi_i(\mathbf{r}) \hat{a}_i$
• The many-body Hamiltonian is expressed using field operators and their derivatives
  • Kinetic energy: $\hat{T} = \int d\mathbf{r} \, \hat{\psi}^\dagger(\mathbf{r}) \left(-\frac{\hbar^2}{2m}\nabla^2\right) \hat{\psi}(\mathbf{r})$
  • Interaction energy: $\hat{V} = \frac{1}{2} \int d\mathbf{r} d\mathbf{r}' \, \hat{\psi}^\dagger(\mathbf{r}) \hat{\psi}^\dagger(\mathbf{r}') V(\mathbf{r}-\mathbf{r}') \hat{\psi}(\mathbf{r}') \hat{\psi}(\mathbf{r})$
• Second quantization simplifies the treatment of indistinguishable particles and symmetrization
• Wick's theorem allows the evaluation of expectation values and correlation functions using normal ordering and contractions

Applications in Quantum Field Theory

• Quantum field theory (QFT) extends the concepts of second quantization to relativistic systems
• Particles are viewed as excitations of underlying quantum fields
  • Each particle type corresponds to a different field (electron field, photon field, etc.)
• Creation and annihilation operators are promoted to field operators satisfying relativistic commutation relations
• The Lagrangian formalism is used to derive equations of motion and conserved quantities
• Feynman diagrams represent perturbative expansions of interaction processes
  • Vertices correspond to interaction terms in the Lagrangian
  • Propagators describe the motion of particles between interactions
• Renormalization techniques handle infinities arising from self-interactions and virtual particles
• QFT provides a framework for describing the Standard Model of particle physics and beyond

Problem-Solving Strategies

• Identify the type of particles involved (bosons or fermions) and the relevant symmetries
• Express the many-body Hamiltonian in terms of creation and annihilation operators
  • Use commutation relations to simplify expressions and derive equations of motion
• Construct the Fock space basis states relevant to the problem
  • Consider the allowed occupation numbers and symmetry constraints
• Apply perturbation theory to treat interactions as small corrections to the non-interacting system
  • Use Wick's theorem to evaluate expectation values and correlation functions
• Exploit conserved quantities and symmetries to simplify calculations
  • Number conservation, momentum conservation, rotational invariance, etc.
• Utilize diagrammatic techniques (Feynman diagrams) to organize and visualize perturbative calculations
• Employ approximation methods when exact solutions are not feasible
  • Mean-field theory, variational methods, Green's functions, etc.
• Verify results by checking limiting cases, dimensional analysis, and comparing with known solutions