First quantization

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:

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

All the properties of the particle could be known using this vector basis.

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.[2] 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.

• Canonical quantization
• Geometric quantization
• Quantization
• Second quantization