Canonical commutation relation

In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, $${\displaystyle [{\hat {x}},{\hat {p}}_{x}]=i\hbar \mathbb {I} }$$

between the position operator x and momentum operator p_x in the x direction of a point particle in one dimension, where [x , p_x] = x p_x − p_x x is the commutator of x and p_x , i is the imaginary unit, and ℏ is the reduced Planck constant h/2π, and ${\displaystyle \mathbb {I} }$ is the unit operator. In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as $${\displaystyle [{\hat {x}}_{i},{\hat {p}}_{j}]=i\hbar \delta _{ij},}$$ where ${\displaystyle \delta _{ij}}$ is the Kronecker delta.

This relation is attributed to Werner Heisenberg, Max Born and Pascual Jordan (1925),^[1][2] who called it a "quantum condition" serving as a postulate of the theory; it was noted by E. Kennard (1927)^[3] to imply the Heisenberg uncertainty principle. The Stone–von Neumann theorem gives a uniqueness result for operators satisfying (an exponentiated form of) the canonical commutation relation.