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.

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:

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