Fock space - Citizendia

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. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. This article assumes some familiarity with Analytic geometry and the concept of a limit. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. A subatomic particle is an elementary or composite Particle smaller than an Atom. It is named for V. A. Fock. Vladimir Aleksandrovich Fock (or Fok, Владимир Александрович Фoк ( December 22 1898 &ndash December 27 1974

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. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. $H$ 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. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. Fock states are the natural basis of this space. A Fock state, in Quantum mechanics, is any state of the Fock space with a well-defined number of particles in each state (See also the Slater determinant. In Quantum mechanics, a Slater determinant is an expression which describes the Wavefunction of a multi- Fermionic system that satisfies anti-symmetry )

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 $H$ . A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system 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). Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another even in principle 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. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 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. A Fock state, in Quantum mechanics, is any state of the Fock space with a well-defined number of particles in each 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. In Physics, an annihilation operator is an Operator that lowers the number of particles in a given state by one 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 $a^{\dagger}(\phi_i)a(\phi_i)$ . In Quantum mechanics, for systems where the total number of particles may not be preserved the number operator is the observable that counts the number of particles

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