Canonical Coordinate

In Physics , the canonical commutation relation is the relation

:

{Link without Title} = i\hbar

among the position

x

and momentum

p

of a point particle in one dimension, where

{Link without Title} =xp-px is the so-called Commutator of

x

and

p

i

is the Imaginary Unit and

\hbar

is the reduced Planck's Constant

h/2\pi

. This relation is attributed to Heisenberg , and it implies his Uncertainty Principle .

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

i\hbar

with

1

:

:

\{x,p\}

= 1

This observation led Dirac to postulate that, in general, the quantum counterparts

\hat f,\hat g

of classical observables

f,g

should satisfy

:

= i\hbar\widehat{\{f,g\}}.\,

In 1927 , Hermann Weyl showed that a literal correspondence between a quantum operator and a classical distribution in Phase Space could not hold. However, he did propose a mechanism, Weyl Quantization , that underlies a mathematical approach to quantization known as Deformation Quantization .

REPRESENTATIONS

According to the standard Mathematical Formulation Of Quantum Mechanics , quantum observables such as

x

and

p

should be represented as Self-adjoint Operator s on some Hilbert Space . It is relatively easy to see that two Operator s 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 Operator s

e^{-ikx}

and

e^{-iap}

. 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

:

{Link without Title} = i\hbar,

valid for the Quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian

{\mathcal L}

. We identify canonical coordinates (such as

x

in the example above, or a field

\phi(x)

in the case of Quantum Field Theory ) and '''canonical momenta'''

\pi_x

(in the example above it is

p

, or more generally, some functions involving the Derivative s of the canonical coordinates with respect to time).

:

\pi_i \equiv rac{\partial {\mathcal L}}{\partial(\partial x_i / \partial t)}

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

:

rac{\partial}{\partial t} \pi_i = rac{\partial {\mathcal L}}{\partial x_i}

The canonical commutation relations then say

:

{Link without Title} = i\hbar\delta_{ij}

where

\delta_{ij}

is the Kronecker Delta .

SEE ALSO

Canonical Quantization

CCR Algebra

Lie Derivative

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Information About

Canonical Coordinate