Fock Space - Example

Example

An example of a pure state of the Fock space is

which describes a collection of particles, one of which has quantum state, another and so on up to the th particle, where each is any state from the single particle Hilbert space . Here juxtaposition is symmetric respectively antisymmetric multiplication in the symmetric and antisymmetric tensor algebra. The general state in a Fock space is a linear combination of pure states. A Fock state that cannot be written as a pure state is called an entangled state.

When we speak of one particle in state , it must be borne in mind that in quantum mechanics identical particles are indistinguishable. In the same Fock space all particles are identical (to describe many species of particles, take the tensor product of as many different Fock spaces as there are species of particles under consideration). It is one of the most powerful features of this formalism that states are implicitly properly symmetrized. For instance, if the above state is fermionic, it will be 0 if two (or more) of the are equal because the anti symmetric (exterior) product . This is a mathematical formulation of the Pauli exclusion principle that no two (or more) fermions can be in the same quantum state. Also, the product of orthonormal states is properly orthonormal by construction (although possibly 0 in the Fermi case when two states are equal).

A useful and convenient basis for a Fock space is the occupancy number basis. Given the choice of a basis of, we can denote the state with particles in state, particles in state, ..., particles in state by

where each takes the value 0 or 1 for fermionic particles and 0, 1, 2, ... for bosonic particles. Such a state is called a Fock state. When the are understood as the steady states of a free field, the Fock states describes an assembly of non-interacting particles in definite numbers. The most general pure state is the linear superposition of Fock states.

Two operators of great importance are the creation and annihilation operators, which upon acting on a Fock state add respectively remove a particle in the ascribed quantum state. They are denoted and respectively, with the quantum state the particle which is "added" by multiplication with respectively "removed" by (even or odd) interior product with which is the adjoint of . It is often convenient to work with states of the basis of so that these operators remove and add exactly one particle in the given basis state. These operators also serve as a basis for more general operators acting on the Fock space, for instance the number operator giving the number of particles in a specific state is .