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A first quantization of a physical system is a semi-classical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well or a bulk electromagnetic field or gravitational field) is treated classically. First quantization is appropriate for studying a single quantum-mechanical system being controlled by a laboratory apparatus that is itself large enough that classical mechanics is applicable to most of the apparatus.
In general, the one-particle state could be described by a complete set of quantum numbers denoted by . For example, the three quantum numbers associated to an electron in a coulomb potential, like the hydrogen atom, form a complete set (ignoring spin). Hence, the state is called and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using . All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state obtaining the completeness relation:
All the properties of the particle could be known using this vector basis.
When turning to N-particle systems, i.e., systems containing N identical particles i.e. particles characterized by the same physical parameters such as mass, charge and spin, an extension of the single-particle state function to the N-particle state function is necessary. A fundamental difference between classical and quantum mechanics concerns the concept of indistinguishability of identical particles. Only two species of particles are thus possible in quantum physics, the so-called bosons and fermions which obey the rules:
Where we have interchanged two coordinates of the state function. The usual wave function is obtained using the Slater determinant and the identical particles theory. Using this basis, it is possible to solve any many-particle problem.[dubious ]