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
ν
. 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
|
ν
⟩
and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using
ψ
ν
(
r
)
=
⟨
r
|
ν
⟩
. All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state
|
ψ
⟩
=
∑
ν
|
ν
⟩
⟨
ν
|
ψ
⟩
obtaining the completeness relation:
∑
ν
|
ν
⟩
⟨
ν
|
=
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, is necessary an extension of single-particle state function
ψ
(
r
)
to the N-particle state function
ψ
(
r
1
,
r
2
,
.
.
.
,
r
N
)
. 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:
ψ
(
r
1
,
.
.
.
,
r
j
,
.
.
.
,
r
k
,
.
.
.
,
r
N
)
=
+
ψ
(
r
1
,
.
.
.
,
r
k
,
.
.
.
,
r
j
,
.
.
.
,
r
N
)
(bosons),
ψ
(
r
1
,
.
.
.
,
r
j
,
.
.
.
,
r
k
,
.
.
.
,
r
N
)
=
−
ψ
(
r
1
,
.
.
.
,
r
k
,
.
.
.
,
r
j
,
.
.
.
,
r
N
)
(fermions).
Where we have interchanged two coordinates
(
r
j
,
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.‹See TfD›
