| Fock Space |
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| CATEGORIES ABOUT FOCK SPACE | |
| quantum mechanics | |
| quantum field theory | |
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Technically, the Fock space is the Hilbert space made from the Direct Sum of Tensor Product s of single-particle Hilbert spaces: : where ''Sν'' 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 State s 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 State s are the natural basis of this space. (See also the Slater Determinant .) EXAMPLE An example of a state of the Fock space is | ||
| U | \phi_1,\phi_2,\cdots,\phi_n
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| Describing ''n'' Particles, One Of Which Has | "http://wwwinformationdelightinfo/encyclopedia/entry/wavefunction" class="copylinks">Wavefunction ''&phi''<sub>1</sub>, another ''&phi''<sub>2</sub> and so on up to the ''n''<sup>th</sup> particle, where each ''&phi''<sub>''i''</sub> is ''any'' wavefunction from the single particle Hilbert space ''H'' When we speak of ''one particle in state &phi<sub>i</sub>'' it must be born in mind that in quantum mechanics identical particles are Indistinguishable , and in a same Fock space all particles are identical (to describe many species of particles, made the tensor products of as many different Fock spaces) 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''><sub>-</sub> is fermionic, it will be 0 if two (or more) of the ''&phi<sub>i</sub>'' 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 |
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| Such A State Is Called A | "http://wwwinformationdelightinfo/encyclopedia/entry/Fock_state" class="copylinks">Fock State Since ''&psi<sub>i</sub>''> are understood as the steady states of the free field, ie, 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 |
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| Two Operators Of Paramount Importance Are The | "http://wwwinformationdelightinfo/encyclopedia/entry/annihilation_and_creation_operators" class="copylinks">Annihilation And Creation Operators , which upon acting on a Fock state respectively remove and add a particle, in the ascribed quantum state They are denoted <math>a(\phi)</math> and <math>a^{\dagger}(\phi)</math> respectively, with ''&phi'' referring to the quantum state ''&phi''> 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 operator 'number of particle in state ''&phi''> is <math>a^{\dagger}(\phi)a(\phi)</math>) |
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