Editing Canonical quantization (section)

==First quantization==
{{Main|First quantization}}

===Single particle systems===

The following exposition is based on [[Paul Dirac|Dirac's]] treatise on quantum mechanics.<ref name="dirac"/>
In the [[classical mechanics]] of a particle, there are dynamic variables which are called coordinates ({{mvar|x}}) and momenta ({{mvar|p}}). These specify the ''state'' of a classical system. The '''canonical structure''' (also known as the [[Symplectic geometry|symplectic]] structure) of [[classical mechanics]] consists of [[Poisson bracket]]s enclosing these variables, such as {{math|1= {''x'', ''p''} = 1}}.  All transformations of variables which preserve  these brackets are allowed as [[canonical transformation]]s in classical mechanics. Motion itself is such a canonical transformation.

By contrast, in [[quantum mechanics]], all significant features of a particle are contained in  a '''state'''  <math>|\psi\rangle</math>, called a [[quantum state]]. Observables are represented by '''operators''' acting on a [[Hilbert space]] of such [[quantum states]].

The eigenvalue of an operator acting on one of its eigenstates represents the value of a measurement on the particle thus represented. For example, the [[energy]] is read off by the [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator <math>\hat{H}</math> acting on  a state <math>|\psi_n\rangle</math>, yielding
<math display="block">\hat{H}|\psi_n\rangle=E_n|\psi_n\rangle,</math>
where {{math|''E<sub>n</sub>''}} is the characteristic energy associated to this <math>|\psi_n\rangle</math> [[eigenstate]].

Any state could be represented as a [[linear combination]] of eigenstates of energy; for example,
<math display="block">|\psi\rangle=\sum_{n=0}^{\infty} a_n |\psi_n\rangle ,</math>where {{math|''a<sub>n</sub>''}} are constant coefficients.

As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones, <math>\hat{X}</math> and <math>\hat{P}</math>, respectively. The connection between this representation and the more usual [[wavefunction]] representation is given by the eigenstate of the position operator <math>\hat{X}</math> representing a particle at position <math>x</math>, which is  denoted by an element <math>|x\rangle</math> in the Hilbert space, and which satisfies <math>\hat{X}|x\rangle = x|x\rangle</math>. Then, <math>\psi(x)= \langle x|\psi\rangle</math>.

Likewise, the eigenstates <math>|p\rangle</math> of  the momentum operator <math>\hat{P}</math> specify the [[Position and momentum space|momentum representation]]: <math>\psi(p)= \langle p|\psi\rangle</math>.

The central relation between these operators is a quantum  analog  of the above [[Poisson bracket]] of classical mechanics, the  '''[[canonical commutation relation]]''',
<math display="block">[\hat{X},\hat{P}] = \hat{X}\hat{P}-\hat{P}\hat{X} = i\hbar.</math>

This relation encodes (and formally leads to) the [[uncertainty principle]], in the form  {{math|Δ''x'' Δ''p'' ≥ ''ħ''/2}}.  This algebraic structure may be thus considered as the quantum analog of the ''canonical structure'' of classical mechanics.

===Many-particle systems===

When turning to N-particle systems, i.e., systems containing N [[identical particles]] (particles characterized by the same [[quantum numbers]] such as [[mass]], [[Electric charge|charge]] and [[Spin (physics)|spin]]), it is necessary to extend the single-particle state function <math>\psi(\mathbf{r})</math> to the N-particle state function <math>\psi(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N)</math>. A fundamental difference between classical and quantum mechanics concerns the concept of [[Identical particles|indistinguishability]] of identical particles. Only two species of particles are thus possible in quantum physics, the so-called [[bosons]] and [[fermions]] which obey the following rules for each kind of particle:

* for bosons: <math display="block">\psi(\mathbf{r}_1,\dots,\mathbf{r}_j,\dots,\mathbf{r}_k,\dots,\mathbf{r}_N)=+\psi(\mathbf{r}_1,\dots,\mathbf{r}_k,\dots,\mathbf{r}_j,\dots,\mathbf{r}_N),</math>
* for fermions: <math display="block">\psi(\mathbf{r}_1,\dots,\mathbf{r}_j,\dots,\mathbf{r}_k,\dots,\mathbf{r}_N)=-\psi(\mathbf{r}_1,\dots,\mathbf{r}_k,\dots,\mathbf{r}_j,\dots,\mathbf{r}_N),</math>

where we have interchanged two coordinates <math>(\mathbf{r}_j, \mathbf{r}_k)</math> 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 various many-particle problems.