Relation satisfied by conjugate variables in quantum mechanics
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,
between the position operator x and momentum operator px in the x direction of a point particle in one dimension, where [x , px] = xpx − pxx is the commutator of x and px, i is the imaginary unit, and ℏ is the reduced Planck constant h/2π, and 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
where 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.
^"The Development of Quantum Mechanics".
^Born, M.; Jordan, P. (1925). "Zur Quantenmechanik". Zeitschrift für Physik. 34 (1): 858–888. Bibcode:1925ZPhy...34..858B. doi:10.1007/BF01328531. S2CID 186114542.
^Kennard, E. H. (1927). "Zur Quantenmechanik einfacher Bewegungstypen". Zeitschrift für Physik. 44 (4–5): 326–352. Bibcode:1927ZPhy...44..326K. doi:10.1007/BF01391200. S2CID 121626384.
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In quantum mechanics, the canonicalcommutationrelation is the fundamental relation between canonical conjugate quantities (quantities which are related...
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in the position representation. Note that the definition above is the canonical momentum, which is not gauge invariant and not a measurable physical quantity...
Position–linear momentum uncertainty relation: for the position and linear momentum operators, the canonicalcommutationrelation [ x ^ , p ^ ] = i ℏ {\displaystyle...
a classical field and {,} is the Peierls bracket. Then, the canonicalcommutationrelation is [ ϕ [ f ] , ϕ [ g ] ] = i Δ [ f , g ] {\displaystyle [\phi...
transformation is an isomorphism of either the canonicalcommutationrelation algebra or canonical anticommutation relation algebra. This induces an autoequivalence...
{\displaystyle {\hat {P}}} do not commute, but rather satisfy the canonicalcommutationrelation: [ X ^ , P ^ ] = i ℏ . {\displaystyle [{\hat {X}},{\hat {P}}]=i\hbar...
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"time" in this relation, because energy and time (unlike position q and momentum p, for example) do not satisfy a canonicalcommutationrelation (such as [q...
Assuming that observables of the coordinate and momentum obey the canonicalcommutationrelation [x̂, p̂] = iħ. Setting H ^ = H ( x ^ , p ^ ) {\displaystyle...
following commutators can be easily obtained by substituting in the canonicalcommutationrelation: [ b k , b k ′ † ] = δ k , k ′ , [ b k , b k ′ ] = [ b k † ...
{\displaystyle {\widehat {p\,}}} , which necessarily satisfy the canonicalcommutationrelation: [ x ^ , p ^ ] = x ^ p ^ − p ^ x ^ = i ℏ {\displaystyle [{\widehat...
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]}+{\mathfrak {U}}({\hat {x}},t)\,.} These operators obey the canonicalcommutationrelation [ x ^ i , p ^ j ± ] = ∓ α ℏ δ j i . {\displaystyle [{\hat {x}}^{i}...
noncommutative. One commonly studied version of such theories has the "canonical" commutationrelation: [ x μ , x ν ] = i θ μ ν {\displaystyle [x^{\mu },x^{\nu }]=i\theta...
{\hat {p}}_{1},\ldots ,{\hat {p}}_{n})^{\mathrm {T} }.} The canonicalcommutationrelation can be expressed simply as [ z ^ , z ^ T ] = i ℏ Ω {\displaystyle...
contributed much to the mathematical form of matrix mechanics, and developed canonical anticommutation relations for fermions. He introduced Jordan algebras...
{\frac {d}{dq}}q-q{\frac {d}{dq}}=1,} coinciding with the usual canonicalcommutationrelation − i [ q , p ] = 1 {\displaystyle -i[q,p]=1} , in position space...
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Global Information![{\displaystyle [{\hat {x}},{\hat {p}}_{x}]=i\hbar \mathbb {I} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2a5584c5b377f65afb20dcff0b5254dab3c77e)
![{\displaystyle [{\hat {x}}_{i},{\hat {p}}_{j}]=i\hbar \delta _{ij},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1d4955c83c1406fda6c02343f967d734433136)
