# 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 $\nu$ . 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 $|\nu \rangle$  and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using $\psi _{\nu }(\mathbf {r} )=\langle \mathbf {r} |\nu \rangle$ . All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state $|\psi \rangle =\sum _{\nu }|\nu \rangle \langle \nu |\psi \rangle$  obtaining the completeness relation:

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

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 $\psi (\mathbf {r} )$  to the N-particle state function $\psi (\mathbf {r} _{1},\mathbf {r} _{2},...,\mathbf {r} _{N})$  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:

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

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

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