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Concept
Fock space
Summary
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space H. It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung" ("Configuration space and second quantization").
Informally, a Fock space is the sum of a set of Hilbert spaces representing zero particle states, one particle states, two particle states, and so on. If the identical particles are bosons, the n-particle states are vectors in a symmetrized tensor product of n single-particle Hilbert spaces H. If the identical particles are fermions, the n-particle states are vectors in an antisymmetrized tensor product of n single-particle Hilbert spaces H (see symmetric algebra and exterior algebra respectively). A general state in Fock space is a linear combination of n-particle states, one for each n.
Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H,
Here is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic or fermionic statistics, and the overline represents the completion of the space. The bosonic (resp. fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) the symmetric tensors (resp. alternating tensors ). For every basis for H there is a natural basis of the Fock space, the Fock states.
The Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space
Here , the complex scalars, consists of the states corresponding to no particles, the states of one particle, the states of two identical particles etc.
A general state in is given by
where
is a vector of length 1 called the vacuum state and is a complex coefficient,
is a state in the single particle Hilbert space and is a complex coefficient,
and is a complex coefficient, etc.
Official source
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