10.1 Fock space and occupation number representation

Fock space and occupation number representation are powerful tools for describing multi-particle quantum systems. They allow us to handle systems with varying particle numbers and provide a compact way to represent complex quantum states.

These concepts are crucial for understanding second quantization in quantum mechanics. By using creation and annihilation operators, we can easily manipulate particle states and study many-body systems more efficiently.

Fock Space for Multi-Particle Systems

Hilbert Space for Describing Quantum States

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• Fock space is a Hilbert space used to describe the quantum states of variable or unknown numbers of identical particles
• It allows for the description of systems with a variable number of particles, such as in quantum field theory and many-body physics
• The dimension of Fock space grows exponentially with the number of available single-particle states, making it an infinite-dimensional space for systems with an unbounded number of particles (e.g., quantum harmonic oscillator, quantum field theory)

Basis States and Operators

• The basis states of Fock space are occupation number states, which specify the number of particles occupying each single-particle state
• Creation and annihilation operators are used to add or remove particles from specific single-particle states in Fock space
  • Creation operators ($a^{\dagger}$) increase the occupation number of a specific single-particle state by one
  • Annihilation operators ($a$) decrease the occupation number of a specific single-particle state by one
• The vacuum state is a special state in Fock space representing the absence of particles in all single-particle states (i.e., all occupation numbers are zero)

Occupation Number Representation

Describing Multi-Particle States

• The occupation number representation uses a string of numbers to specify the number of particles occupying each single-particle state
• Each single-particle state is assigned an index or quantum number, and the occupation number for that state represents the number of particles in that state
• The occupation numbers are non-negative integers, with 0 indicating an unoccupied state and positive integers representing the number of particles in the state (e.g., $|1,0,2\rangle$ represents 1 particle in the first state, 0 in the second, and 2 in the third)
• The total number of particles in the system is given by the sum of the occupation numbers across all single-particle states

Compact Representation and Operators

• The occupation number representation allows for a compact description of multi-particle states without explicitly writing out the wave function
• Operators in the occupation number representation act on the occupation numbers, changing the number of particles in specific states
  • Creation operators increase the occupation number of a specific state by one (e.g., $a^{\dagger}_1|1,0,2\rangle = \sqrt{2}|2,0,2\rangle$)
  • Annihilation operators decrease the occupation number of a specific state by one (e.g., $a_3|1,0,2\rangle = \sqrt{2}|1,0,1\rangle$)

Basis States Interpretation

Physical Meaning of Basis States

• Each basis state in the occupation number representation corresponds to a specific configuration of particles distributed among the available single-particle states
• The occupation numbers in a basis state provide information about the number of particles in each single-particle state (e.g., $|1,0,2\rangle$ represents 1 particle in the first state, 0 in the second, and 2 in the third)
• Basis states with different occupation numbers represent distinct physical configurations of the multi-particle system

Energy and Orthogonality

• The energy of a basis state depends on the occupation numbers and the energies of the corresponding single-particle states
  • The total energy is the sum of the energies of the occupied single-particle states (e.g., $E = n_1\epsilon_1 + n_2\epsilon_2 + ...$, where $n_i$ is the occupation number and $\epsilon_i$ is the energy of the $i$-th state)
• Basis states with the same total number of particles but different occupation number distributions are orthogonal to each other (i.e., their inner product is zero)

Boson vs Fermion Wave Functions

Symmetry Properties under Particle Exchange

• Bosons and fermions are two classes of particles with different symmetry properties under particle exchange
• The multi-particle wave function for bosons is symmetric under the exchange of any two particles, meaning it remains unchanged (e.g., $\psi(x_1, x_2) = \psi(x_2, x_1)$ for bosons)
• The multi-particle wave function for fermions is antisymmetric under the exchange of any two particles, meaning it changes sign (e.g., $\psi(x_1, x_2) = -\psi(x_2, x_1)$ for fermions)

Occupation Numbers and Pauli Exclusion Principle

• Bosons can occupy the same single-particle state, leading to the possibility of multiple bosons in the same state (occupation numbers greater than 1)
  • Examples of bosons include photons, gluons, and certain atomic nuclei (e.g., helium-4)
• Fermions obey the Pauli exclusion principle, which states that no two fermions can occupy the same single-particle state simultaneously (occupation numbers limited to 0 or 1)
  • Examples of fermions include electrons, protons, neutrons, and quarks
• The symmetry properties of the wave function have consequences for the allowed occupation number configurations and the resulting physical properties of the system (e.g., Bose-Einstein condensation for bosons, Fermi-Dirac statistics for fermions)
• The symmetrization or antisymmetrization of the wave function ensures the proper behavior of bosons and fermions in multi-particle systems