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.
One-particle systems
In general, the one-particle state could be described by a complete set of quantum numbers denoted by ν . For example, the three quantum numbers n , l , m associated to an electron in a coulomb potential, like the hydrogen atom, form a complete set (ignoring spin). Hence, the state is called | ν ⟩ and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using ψ ν ( r ) = ⟨ r | ν ⟩ . All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state | ψ ⟩ = ∑ ν | ν ⟩ ⟨ ν | ψ ⟩ obtaining the completeness relation:
∑ ν | ν ⟩ ⟨ ν | = 1 ^
All the properties of the particle could be known using this vector basis.
Many-particle systems
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, is necessary an extension of single-particle state function ψ ( r ) to the N-particle state function ψ ( r 1 , r 2 , . . . , r N ) . 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:
ψ ( r 1 , . . . , r j , . . . , r k , . . . , r N ) = + ψ ( r 1 , . . . , r k , . . . , r j , . . . , r N ) (bosons),
ψ ( r 1 , . . . , r j , . . . , r k , . . . , r N ) = − ψ ( r 1 , . . . , r k , . . . , r j , . . . , r N ) (fermions).
Where we have interchanged two coordinates ( r j , r k ) 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.‹See TfD›
