Editing Weyl algebra (section)

== Motivation ==
{{see also|Canonical commutation relation}}
The Weyl algebra arises naturally in the context of [[quantum mechanics]] and the process of [[canonical quantization]]. Consider a classical [[phase space]] with canonical coordinates <math>(q_1, p_1, \dots, q_n, p_n) </math>.  These coordinates satisfy the [[Poisson bracket]] relations:<math display="block">
\{q_i, q_j\} = 0, \quad \{p_i, p_j\} = 0, \quad \{q_i, p_j\} = \delta_{ij}.
</math>In canonical quantization, one seeks to construct a [[Hilbert space]] of states and represent the classical observables (functions on phase space) as [[Self-adjoint operator|self-adjoint operators]] on this space. The canonical commutation relations are imposed:<math display="block">
[\hat{q}_i, \hat{q}_j] = 0, \quad [\hat{p}_i, \hat{p}_j] = 0, \quad [\hat{q}_i, \hat{p}_j] = i\hbar \delta_{ij},
</math>where <math>[\cdot, \cdot]</math> denotes the [[commutator]]. Here, <math>\hat{q}_i</math> and <math>\hat{p}_i</math> are the operators corresponding to <math>q_i</math> and <math>p_i</math> respectively. [[Erwin Schrödinger]] proposed in 1926 the following:{{sfn | Landsman | 2007 | p=428}}

* <math>\hat{q_j}</math> with multiplication by <math>x_j</math>.
* <math>\hat{p}_j</math> with <math>-i\hbar \partial_{x_j}</math>.

With this identification, the canonical commutation relation holds.