In quantum mechanics, the Fock representation, also known as the second quantization, is a mathematical framework used to describe systems with an arbitrary number of identical particles. It provides a convenient way to represent the states of particles in terms of occupation numbers.
In the Fock representation, the state of a system is described by a ket vector, which is a linear combination of basis states. Each basis state corresponds to a different configuration of particles occupying different quantum states. The occupation numbers specify how many particles are in each quantum state.
For example, consider a system of bosons, which are particles that obey Bose-Einstein statistics. The Fock space associated with this system consists of basis states labeled by occupation numbers. The occupation number representation can be denoted as |n₁, n₂, n₃, ...⟩, where n₁, n₂, n₃, ... represent the number of particles in the corresponding quantum states.
The Fock space basis states are constructed by applying creation operators to a reference state, usually denoted as the vacuum state |0⟩, which represents the absence of particles. The creation operator aᴶ⁺ creates a particle in the quantum state labeled by the index J. The ket vector representing a state with a specific occupation is given by:
|n₁, n₂, n₃, ...⟩ = (a₁ᴶ⁺)^(n₁) (a₂ᴶ⁺)^(n₂) (a₃ᴶ⁺)^(n₃) ... |0⟩
In the Fock representation, the operators that describe the dynamics of the system, such as the Hamiltonian or annihilation operators, are expressed in terms of creation and annihilation operators acting on the basis states.
The Fock representation is particularly useful when dealing with systems where the number of particles can vary, such as in quantum field theory or many-body quantum systems. It allows for a concise and elegant formulation of quantum mechanics for such systems, where the individual particles' identities are not distinguished, and only the overall occupation numbers matter.
It's worth noting that the Fock representation is distinct from the position or momentum representations typically used to describe single-particle quantum mechanics, where the states are represented as wavefunctions in continuous variables. In contrast, the Fock representation deals with states described by occupation numbers in discrete quantum states.