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| In [[quantum mechanics]] ([[physics]]), 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,
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| :<math>[x,p_x] = i\hbar</math>
| | To conclude, clash of clans hack tool no article must not be enabled to get in the way of the bigger question: what makes we above? Putting this particular aside its of great signification. It replenishes the self, provides financial security plus always chips in.<br><br>clash of clans is a ideal game, which usually requires in order to build your personal village, discover warriors, raid aspects and build your extremely clan and so 4th. there is a lot a lot good deal to this video game and for every one too you require jewels within order to play, as you really like. Clash of Clans hack allows you to obtain as many jewels as you wish. There is an unlimited volume gems you could yield with all the Clash of Clans cheats possible online, however you need to be specific about the web page link you are using for some of them but waste materials your time and also dont get anybody anything more.<br><br>Game applications are fun, nonetheless typically also be costly. The costs of game titles and consoles can be more expensive than many people would choose those to be, but this may sometimes eliminated.<br><br>Make sure your child's xbox play enjoying. Video computer game are now rated typically like films and generally can help. That enables you to keep an eye on the information your kids will be exposed to. Depending upon your child's age, continue to keep her clear of video online video media that happen to prove meant for people what individuals are more fully produced than him.<br><br>All of this construction is what options that you can you should be a part of a clan, however it likewise houses reinforcement troops. Click a button to assist you to ask your clan in order to really send you some troops, and they are choosing to be out there to make use off in assaults, or to finally defend your base to work with you while you're using your weekly LARPing group. Upgrading this constructing permits extra troops to positively be stored for defenders. If you enjoyed this write-up and you would such as to receive even more information regarding [http://circuspartypanama.com clash of clans cheats iphone] kindly go to the webpage. You may are in need of 20 available slots so that it will get a dragon. This is a wonderful base for players trying to shield trophies in addition to never worried about extracts. Players will retrieve it hard to wash out your city corridor. Most will resolve for the easy get and take out a person's assets.<br><br>Basically, it would alone acquiesce all of us for tune 2 volume points. If you appetite for you to songs added than in which - as Supercell finely acquainted t had actually been all-important - you guarantees assorted beeline segments. Theoretically they could race track alike added bulk items. If they capital to help allegation added or beneath for a two day skip, they are able to calmly familiarize 1 included in segment. |
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| between the position {{mvar|x}} and momentum {{mvar|p<sub>x</sub>}} in the {{mvar|x}} direction of a point particle in one dimension, where {{math| [''x'' , ''p''<sub>''x''</sub>] {{=}} ''x'' ''p''<sub>''x''</sub> − ''p''<sub>''x''</sub> ''x''}} is the [[commutator]] of {{mvar|x}} and {{mvar|p<sub>x</sub> }}, {{mvar|i}} is the [[imaginary unit]], and {{math|ℏ}} is the reduced [[Planck's constant]] {{math|{{sfrac|''h''|2π}}}} .
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| This relation is attributed to [[Max Born]] (1925),<ref>{{cite doi|10.1007/BF01328531|noedit}}</ref> who called it a "quantum condition" serving as a postulate of the theory; it was noted by [[Earle Hesse Kennard|E. Kennard]] (1927)<ref>{{cite doi|10.1007/BF01391200|noedit}}</ref> to imply the [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]].
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| == Relation to classical mechanics ==
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| 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]] multiplied by {{math|''i''ℏ}}:
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| :<math>\{x,p\} = 1 \, .</math>
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| This observation led [[Paul Dirac|Dirac]] to propose that the quantum counterparts {{mvar|f̂}}, {{mvar|ĝ}} of classical observables {{mvar|f}}, {{mvar|g}} satisfy
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| :<math>[\hat f,\hat g]= i\hbar\widehat{\{f,g\}} \, .</math>
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| In 1946, [[Hilbrand J. Groenewold|Hip Groenewold]] demonstrated that a ''general systematic correspondence'' between quantum commutators and Poisson brackets could not hold consistently. <ref name="groenewold">{{cite doi|10.1016/S0031-8914(46)80059-4|noedit}}</ref> However, he did appreciate that such a systematic correspondence does, in fact, exist between the quantum commutator and a ''deformation'' of the Poisson bracket, the [[Moyal bracket]], and, in general, quantum operators and classical observables and distributions in [[phase space]]. He thus finally elucidated the correspondence mechanism, [[Weyl quantization]], that underlies an alternate equivalent mathematical approach to quantization known as [[deformation quantization]].<ref name="groenewold"/>
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| == Representations ==
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| The [[Lie group|group]] {{math|''H''<sub>3</sub>(ℝ)}} generated by [[exponential map|exponentiation]] of the [[Lie algebra]] specified by these commutation relations, {{math|1=['''x''', '''p'''] = ''i''ℏ}}, is called the [[Heisenberg group]].
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| According to the standard [[mathematical formulation of quantum mechanics]], quantum observables such as {{math|'''x'''}} and {{math|'''p'''}} should be represented as [[self-adjoint operator]]s on some [[Hilbert space]]. It is relatively easy to see that two [[operator (mathematics)|operator]]s satisfying the above canonical commutation relations cannot both be [[bounded operator|bounded]]—try taking the [[trace (linear algebra)|Trace]] of both sides of the relations and use the relation {{math|1=Trace(''A B '') = Trace(''B A '')}}; one gets a finite number on the right and zero on the left.<ref>More directly, note {{math|1=[''x<sup>n</sup>'', ''p''] = ''i'' ℏ ''n x''<sup>''n'' − 1</sup>}}, hence {{math|<big>2 ‖ ''p'' ‖ ‖ ''x'' ‖<sup>''n''</sup> ≥ ''n'' ℏ ‖ ''x'' ‖<sup>''n'' − 1</sup></big>}}, so that, {{math|∀''n'': 2 ‖ ''p'' ‖ ‖ ''x'' ‖ ≥ ''n'' ℏ}}. However, {{mvar|n}} can be arbitrarily large. Utilizing the Weyl relations, below, it can actually be shown that ''both'' operators are unbounded.</ref>
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| These canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded) [[unitary operator]]s {{math|exp(''i t'' '''x''')}} and {{math|exp(''i s'' '''p''')}}, which do admit finite-dimensional representations. The resulting braiding relations for these are the so-called [[Stone–von Neumann theorem|Weyl relations]]
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| :{{math|size=120%|1=exp(''i t'' '''x''') exp(''i s'' '''p''') = exp(−''i''ℏ ''s t'') exp(''i s'' '''p''') exp(''i t'' '''x''')}}.
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| The corresponding group [[commutator]] is then
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| :{{math|size=120%|1=exp(''i t'' '''x''') exp(''i s'' '''p''') exp(−''i t'' '''x''') exp(−''i s'' '''p''') = exp(−''i''ℏ ''s t'')}}.
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| The uniqueness of the canonical commutation relations between position and momentum is then guaranteed by the [[Stone–von Neumann theorem]].
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| == Generalizations ==
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| The simple formula
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| :<math>[x,p] = i\hbar, \, </math>
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| valid for the [[Canonical quantization|quantization]] of the simplest classical system, can be generalized to the case of an arbitrary [[Lagrangian]] <math>{\mathcal L}</math>.<ref name="town">{{cite book |first=J. S. |last=Townsend |title=A Modern Approach to Quantum Mechanics |publisher=University Science Books |location=Sausalito, CA |year=2000 |isbn=1-891389-13-0 }}</ref> We identify '''canonical coordinates''' (such as {{mvar|x}} in the example above, or a field {{math|φ(''x'')}} in the case of [[quantum field theory]]) and '''canonical momenta''' {{math|π<sub>''x''</sub>}} (in the example above it is {{mvar|p}}, or more generally, some functions involving the [[derivative]]s of the canonical coordinates with respect to time):
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| :<math>\pi_i \ \stackrel{\mathrm{def}}{=}\ \frac{\partial {\mathcal L}}{\partial(\partial x_i / \partial t)}.</math>
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| This definition of the canonical momentum ensures that one of the [[Euler–Lagrange equation]]s has the form
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| :<math>\frac{\partial}{\partial t} \pi_i = \frac{\partial {\mathcal L}}{\partial x_i}.</math>
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| The canonical commutation relations then amount to
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| :<math>[x_i,\pi_j] = i\hbar\delta_{ij}, \, </math>
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| where {{math|''δ''<sub>''ij''</sub>}} is the [[Kronecker delta]].
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| Further, it can be easily shown that
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| :<math>[p_i,F(\vec{x})] = -i\hbar\frac{\partial F(\vec{x})}{\partial x_i}; \qquad [x_i, F(\vec{p})] = i\hbar\frac{\partial F(\vec{p})}{\partial p_i}.</math>
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| ==Gauge invariance==
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| Canonical quantization is applied, by definition, on [[canonical coordinates]]. However, in the presence of an [[electromagnetic field]], the canonical momentum {{mvar|p}} is not [[gauge invariant]]. The correct gauge-invariant momentum (or "kinetic momentum") is
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| :<math>p_\textrm{kin} = p - qA \,\!</math> ([[SI units]]) <math>p_\textrm{kin} = p - \frac{qA}{c} \,\!</math> ([[Gaussian units|cgs units]]),
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| where {{mvar|q}} is the particle's [[electric charge]], {{mvar|A}} is the [[magnetic potential|vector potential]], and {{math|''c''}} is the [[speed of light]]. Although the quantity {{math|''p''<sub>kin</sub>}} is the "physical momentum", in that it is the quantity to be identified with momentum in laboratory experiments, it ''does not'' satisfy the canonical commutation relations; only the canonical momentum does that. This can be seen as follows.
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| The non-relativistic [[Hamiltonian (quantum mechanics)|Hamiltonian]] for a quantized charged particle of mass {{mvar|m}} in a classical electromagnetic field is (in cgs units)
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| :<math>H=\frac{1}{2m} \left(p-\frac{qA}{c}\right)^2 +q\phi</math>
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| where {{mvar|A}} is the three-vector potential and <math>\phi</math> is the [[scalar potential]]. This form of the Hamiltonian, as well as the [[Schrödinger equation]] <math>H\psi=i\hbar \partial\psi/\partial t</math>, the [[Maxwell equation]]s and the [[Lorentz force law]] are invariant under the gauge transformation
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| :<math>A\to A^\prime=A+\nabla \Lambda</math>
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| :<math>\phi\to \phi^\prime=\phi-\frac{1}{c} \frac{\partial \Lambda}{\partial t}</math>
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| :<math>\psi\to\psi^\prime=U\psi</math>
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| :<math>H\to H^\prime= U HU^\dagger,</math>
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| where
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| :<math>U=\exp \left( \frac{iq\Lambda}{\hbar c}\right)</math>
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| and <math>\Lambda=\Lambda(x,t)</math> is the gauge function.
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| The [[angular momentum operator]] is
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| :<math>L=r \times p \,\!</math>
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| and obeys the canonical quantization relations
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| :<math>[L_i, L_j]= i\hbar {\epsilon_{ijk}} L_k</math>
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| defining the [[Lie algebra]] for [[so(3)]], where <math>\epsilon_{ijk}</math> is the [[Levi-Civita symbol]]. Under gauge transformations, the angular momentum transforms as
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| :<math> \langle \psi \vert L \vert \psi \rangle \to
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| \langle \psi^\prime \vert L^\prime \vert \psi^\prime \rangle =
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| \langle \psi \vert L \vert \psi \rangle +
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| \frac {q}{\hbar c} \langle \psi \vert r \times \nabla \Lambda \vert \psi \rangle \, .
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| </math>
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| The gauge-invariant angular momentum (or "kinetic angular momentum") is given by
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| :<math>K=r \times \left(p-\frac{qA}{c}\right),</math> | |
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| which has the commutation relations
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| :<math>[K_i,K_j]=i\hbar {\epsilon_{ij}}^{\,k}
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| \left(K_k+\frac{q\hbar}{c} x_k
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| \left(x \cdot B\right)\right)</math>
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| where
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| :<math>B=\nabla \times A</math>
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| is the [[magnetic field]]. The inequivalence of these two formulations shows up in the [[Zeeman effect]] and the [[Aharonov–Bohm effect]].
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| ==Angular momentum operators==
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| From {{math|1=''L''<sub>''x''</sub> = ''y p<sub>z</sub>'' − ''z p<sub>y</sub>''}}, etc, it follows directly from the above that
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| :<math> [{L_x}, {L_y}] = i \hbar \epsilon_{xyz} {L_z}, </math>
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| where <math>\epsilon_{xyz}</math> is the [[Levi-Civita symbol]] and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for the [[spin (physics)|spin]] operators.
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| All such nontrivial commutation relations for pairs of operators lead to corresponding [[uncertainty principle|uncertainty relations]],<ref name="robertson">{{cite journal |first=H. P. |last=Robertson |title=The Uncertainty Principle |journal=[[Physical Review]] |volume=34 |issue=1 |year=1929 |pages=163–164 |doi=10.1103/PhysRev.34.163 |bibcode = 1929PhRv...34..163R }}</ref> involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two [[Self-adjoint operator|Hermitian operators]] {{mvar|A}} and {{mvar|B}}, consider expectation values in a system in the state {{mvar|ψ}}, the variances around the corresponding expectation values being {{math|1=(Δ''A'')<sup>2</sub> ≡ {{langle}}(''A'' − {{langle}}''A''{{rangle}})<sup>2</sup>{{rangle}}}}, etc.
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| Then
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| :<math> \Delta A \, \Delta B \geq \frac{1}{2} \sqrt{ \left|\left\langle\left[{A},{B}\right]\right\rangle \right|^2 + \left|\left\langle\left\{ A-\langle A\rangle ,B-\langle B\rangle \right\} \right\rangle \right|^2} ,</math>
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| where {{math|1=[''A'', ''B''] ≡ ''A B'' − ''B A''}} is the [[Commutator#Ring theory|commutator]] of {{mvar|A}} and {{mvar|B}}, and {{math|1={''A'', ''B''} ≡ ''A B'' + ''B A''}} is the [[anticommutator]].
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| This follows through use of the [[Cauchy–Schwarz inequality]], since
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| {{math|{{!}}{{langle}}''A''<sup>2</sup>{{rangle}}{{!}} {{!}}{{langle}}''B''<sup>2</sup>{{rangle}}{{!}} ≥ {{!}}{{langle}}''A B''{{rangle}}{{!}}<sup>2</sup>}}, and {{math|1=''A B'' = ([''A'', ''B''] + {''A'', ''B''})/2 }}; and similarly for the shifted operators {{math|''A'' − {{langle}}''A''{{rangle}}}} and {{math|''B'' − {{langle}}''B''{{rangle}}}}. (cf. [[Uncertainty principle derivations]].)
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| Judicious choices for {{mvar|A}} and {{mvar|B}} yield Heisenberg's familiar uncertainty relation for {{mvar|x}} and {{mvar|p}}, as usual.
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| Here, for {{mvar|L<sub>x</sub>}} and {{mvar|L<sub>y</sub> }},<ref name="robertson" /> in angular momentum multiplets {{math|1=''ψ'' = {{!}}''{{ell}}'',''m''{{rangle}}}}, one has {{math|1={{langle}}''L<sub>x</sub>''<sup>2</sup>{{rangle}} = {{langle}}''L<sub>y</sub>''<sup>2</sup>{{rangle}} = (''{{ell}}'' (''{{ell}}'' + 1) − ''m''<sup>2</sup>) ℏ<sup>2</sup>/2 }}, so the above inequality yields useful constraints such as a lower bound on the [[Casimir invariant]] {{math|''{{ell}}'' (''{{ell}}'' + 1) ≥ ''m'' (''m'' + 1)}}, and hence {{math|''{{ell}}'' ≥ ''m''}}, among others.
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| == See also ==
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| *[[Canonical quantization]]
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| *[[CCR algebra]]
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| *[[Lie derivative]]
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| *[[Moyal bracket]]
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| ==References==
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| {{Reflist}}
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| [[Category:Quantum mechanics]]
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| [[Category:Mathematical physics]]
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| [[zh:對易關係]]<!-- in Chinese wikipedia 對易關係 is redirected to 交換子; once the redirect is canceled, 對易關係 should be added to wikidata. -->
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