Canonical commutation relation

Template:TopicInQuantum-theory In physics, the canonical commutation relation is the relation

[itex][x,p] = i\hbar[itex]

among the position [itex]x[itex] and momentum [itex]p[itex] of a point particle in one dimension, where [itex][x,p]=xp-px[itex] is the so-called commutator of [itex]x[itex] and [itex]p[itex], [itex]i[itex] is the imaginary unit and [itex]\hbar[itex] is the reduced Planck's constant [itex]h/2\pi[itex]. This relation is attributed to Heisenberg, and it implies his uncertainty principle.

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Relation to classical mechanics

By contrast, in classical physics all observables commute and the commutator would be zero; however, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket and the constant [itex]i\hbar[itex] with [itex]1[itex]:

[itex]\{x,p\}[itex] = 1

This observation led Dirac to postulate that, in general, the quantum counterparts [itex]\hat f,\hat g[itex] of classical observables [itex]f,g[itex] should satisfy

[itex][\hat f,\hat g]= i\hbar\widehat{\{f,g\}}.\,[itex]

Representations

According to the standard mathematical formulation of quantum mechanics, quantum observables such as [itex]x[itex] and [itex]p[itex] should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operators satisfying the canonical commutation relations cannot both be bounded. The canonical commutation relations can be made tamer by writing them in terms of the (bounded) unitary operators [itex]e^{-ikx}[itex] and [itex]e^{-iap}[itex]. The result is the so-called Weyl relations. The uniqueness of the canonical commutation relations between position and momentum is guaranteed by the Stone-von Neumann theorem. The group associated with the commutation relations is called the Heisenberg group.

Generalizations

The simple formula

[itex][x,p] = i\hbar[itex],

valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian [itex]{\mathcal L}[itex]. We identify canonical coordinates (such as [itex]x[itex] in the example above, or a field [itex]\phi(x)[itex] in the case of quantum field theory) and canonical momenta [itex]\pi_x[itex] (in the example above it is [itex]p[itex], or more generally, some functions involving the derivatives of the canonical coordinates with respect to time).

[itex]\pi_i \equiv \frac{\partial {\mathcal L}}{(\partial x_i / \partial t)}[itex]

This definition of the canonical momentum ensures that one of the Euler-Lagrange equations has the form

[itex]\frac{\partial}{\partial t} \pi_i = \frac{\partial {\mathcal L}}{\partial x_i}[itex]

The canonical commutation relations then say

[itex][x_i,\pi_j] = i\hbar\delta_{ij}[itex]

where [itex]\delta_{ij}[itex] is the Kronecker symbol.

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