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. 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 ν {displaystyle u } . For example, the three quantum numbers n , l , m {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 | ν ⟩ {displaystyle | u angle } and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using ψ ν ( r ) = ⟨ r | ν ⟩ {displaystyle psi _{ u }(mathbf {r} )=langle mathbf {r} | u angle } . All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state | ψ ⟩ = ∑ ν | ν ⟩ ⟨ ν | ψ ⟩ {displaystyle |psi angle =sum _{ u }| u angle langle u |psi angle } obtaining the completeness relation: ∑ ν | ν ⟩ ⟨ ν | = 1 ^ {displaystyle sum _{ u }| u angle langle u |=mathbf {hat {1}} }