Fock Space and Occupation Numbers
Fock space is the quantum state space for systems with a variable number of identical particles; a state is specified simply by listing how many particles occupy each single-particle mode, the occupation numbers.
Definition
Fock space is the Hilbert space spanned by states of definite occupation number for each single-particle mode, encompassing all particle numbers from the vacuum upward, with symmetric occupations for bosons and occupations restricted to zero or one for fermions.
Scope
The topic covers the construction of Fock space as the direct sum of symmetrized or antisymmetrized many-particle spaces, the vacuum state with no particles, the occupation-number basis labeling states by mode populations, the restriction of fermionic occupation numbers to zero or one, the number operator counting particles in each mode, and the role of Fock space as the arena for second quantization.
Core questions
- How is Fock space built from single-particle states?
- What is the occupation-number representation and why is it convenient?
- How do bosonic and fermionic occupation numbers differ?
- What does the number operator measure in this representation?
Key concepts
- Fock space
- vacuum state
- occupation-number basis
- number operator
- particle-number conservation
- many-body Hilbert space
Key theories
- Occupation-number basis
- Because identical particles are indistinguishable, a many-particle state is fully specified by how many particles sit in each mode, so the natural basis lists occupation numbers built on a vacuum state, automatically respecting the required exchange symmetry.
- Bosonic versus fermionic occupations
- Bosonic modes can hold any number of particles while fermionic modes are limited to zero or one by the exclusion principle, and the number operator for each mode returns its occupation, giving a unified bookkeeping for systems of variable particle number.
Clinical relevance
Fock space is the working setting of many-body and field theory: it describes photons in quantum optics, phonons and electronic excitations in solids, and particle creation in quantum field theory, and the occupation-number picture is how quantum gases and lattice models are formulated and computed.
History
Fock introduced the space named after him in 1932 to handle variable particle number; it grew from Dirac and Jordan's quantization of fields and became the standard framework for many-body physics and quantum field theory.
Key figures
- Vladimir Fock
- Paul Dirac
- Pascual Jordan
- Eugene Wigner
Related topics
Seminal works
- fetterwalecka2003
- sakurai2017
Frequently asked questions
- Why use occupation numbers instead of wavefunctions for many particles?
- Because identical particles cannot be labeled, tracking which particle is where is meaningless; listing only how many particles occupy each mode captures all the physical information and automatically builds in the correct symmetry, greatly simplifying many-body calculations.
- Why are fermionic occupation numbers limited to zero or one?
- The Pauli exclusion principle forbids two identical fermions from sharing a single-particle state, so each fermionic mode can be either empty or singly occupied, unlike bosonic modes which admit any occupation.