Fock space

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The Fock space is an algebraic system (Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named for V. A. Fock.

Technically, the Fock space is the Hilbert space made from the direct sum of tensor products of single-particle Hilbert spaces:

$F_\nu(H)=\bigoplus_{n=0}^{\infty}S_\nu H^{\otimes n}$

where $S ν$ is the operator which symmetrizes or antisymmetrizes the space, depending on whether the Hilbert space describes particles obeying bosonic $(ν = + )$ or fermionic $(ν = - )$ statistics respectively. $H$ is the single particle Hilbert space. It describes the quantum states for a single particle, and to describe the quantum states of systems with $n$ particles, or superpositions of such states, one must use a larger Hilbert space, the Fock space, which contains states for unlimited and variable number of particles. Fock states are the natural basis of this space. (See also the Slater determinant.)

[edit] Example

An example of a state of the Fock space is

$|\Psi\rangle_\nu=|\phi_1,\phi_2,\cdots,\phi_n\rangle_\nu$

describing $n$ particles, one of which has wavefunction $φ 1$ , another $φ 2$ and so on up to the $n$ ^th particle, where each $φ i$ is any wavefunction from the single particle Hilbert space $H$ . When we speak of one particle in state $φ i$ , it must be borne in mind that in quantum mechanics identical particles are indistinguishable, and 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 intrinsically properly symmetrized. So that for instance, if the above state $|\Psi\rangle_-$ is fermionic, it will be 0 if two (or more) of the $φ i$ are equal, because by the Pauli exclusion principle no two (or more) fermions can be in the same quantum state. Also, the states are properly normalized, by construction.

A useful and convenient basis for this space is the occupancy number basis. If $|\Psi_i\rangle$ is a basis of $H$ , then we can agree to denote the state with $n 0$ particles in state $|\Psi_0\rangle$ , $n 1$ particles in state $|\Psi_1\rangle$ , ..., $n k$ particles in state $|\Psi_k\rangle$ by

$|n_0,n_1,\cdots,n_k\rangle_\nu,$

with each $n i$ taking the value 0 or 1 for fermionic particles and 0, 1, 2, ... for bosonic particles.

Such a state is called a Fock state. Since $|\Psi_i\rangle$ are understood as the steady states of the free field, i.e., a definite number of particles, a Fock state 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 paramount importance are the creation and annihilation operators, which upon acting on a Fock state respectively add and remove a particle in the ascribed quantum state. They are denoted $a^{\dagger}(\phi_i)$ and $a(\phi_i)\,$ respectively, with $\phi_i\,$ referring to the quantum state $|\phi_i\rangle$ in which the particle is removed or added. It is often convenient to work with states of the basis of $H$ so that these operators remove and add exactly one particle in the given 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 $|\phi_i\rangle$ is $a^{\dagger}(\phi_i)a(\phi_i)$ .