# First quantization

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 ${\displaystyle \nu }$. For example, the three quantum numbers ${\displaystyle n,l,m}$ associated to an electron in a coulomb potential, like the hydrogen atom, form a complete set (ignoring spin). Hence, the state is called ${\displaystyle |\nu \rangle }$ and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using ${\displaystyle \psi _{\nu }({\mathbf {r}})=\langle {\mathbf {r}}|\nu \rangle }$. All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state ${\displaystyle |\psi \rangle =\sum _{\nu }|\nu \rangle \langle \nu |\psi \rangle }$ obtaining the completeness relation:

${\displaystyle \sum _{\nu }|\nu \rangle \langle \nu |={\mathbf {\hat {1}}}}$

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, an extension of the single-particle state function ${\displaystyle \psi ({\mathbf {r}})}$ to the N-particle state function ${\displaystyle \psi ({\mathbf {r}}_{1},{\mathbf {r}}_{2},...,{\mathbf {r}}_{N})}$ is necessary.[1] 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:

${\displaystyle \psi ({\mathbf {r}}_{1},...,{\mathbf {r}}_{j},...,{\mathbf {r}}_{k},...,{\mathbf {r_{N}}})=+\psi ({\mathbf {r}}_{1},...,{\mathbf {r}}_{k},...,{\mathbf {r}}_{j},...,{\mathbf {r}}_{N})}$ (bosons),

${\displaystyle \psi ({\mathbf {r}}_{1},...,{\mathbf {r}}_{j},...,{\mathbf {r}}_{k},...,{\mathbf {r_{N}}})=-\psi ({\mathbf {r}}_{1},...,{\mathbf {r}}_{k},...,{\mathbf {r}}_{j},...,{\mathbf {r}}_{N})}$ (fermions).

Where we have interchanged two coordinates ${\displaystyle ({\mathbf {r}}_{j},{\mathbf {r}}_{k})}$ 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.[dubious ]