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| '''Zitterbewegung''' (English: "trembling motion", from [[German language|German]]) is a theoretical rapid motion of elementary particles, in particular electrons, that obey the [[Dirac equation]]. The existence of such motion was first proposed by [[Erwin Schrödinger]] in 1930 as a result of his analysis of the [[wave packet]] solutions of the Dirac equation for [[theory of relativity|relativistic]] electrons in free space, in which an [[Interference (wave propagation)#Quantum interference|interference]] between positive and negative [[energy state]]s produces what appears to be a fluctuation (at the speed of light) of the position of an electron around the median, with a circular frequency of <math>2 m c^2 / \hbar \,\!</math>, or approximately 1.6{{e|21}} [[hertz|Hz]]. A re-examination of Dirac theory, however, shows that interference between positive and negative energy states may not be a necessary criterion for observing zitterbewegung.<ref>{{cite journal|last=David Hestenes|title=The zitterbewegung interpretation of quantum mechanics|journal=Foundations of Physics|year=1990|url=http://link.springer.com/article/10.1007%2FBF01889466|volume=20|issue=10}}</ref> | | Hello, I'm Meagan, a 26 year old from Lauf, Germany.<br>My hobbies include (but are not limited to) Gongoozling, Association football and watching How I Met Your Mother.<br><br>My web-site: [http://Eddies-Nl.nl/forum/profile.php?id=97420 Fifa 15 Coin Generator] |
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| Zitterbewegung of a free relativistic particle has never been observed, but the behavior of such a particle has been ''simulated'' with a trapped ion, by putting it in an environment such that the non-relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac equation (although the physical situation is different).<ref>{{cite news|title=Quantum physics: Trapped ion set to quiver|url=http://www.nature.com/nature/journal/v463/n7277/full/463037a.html|newspaper=Nature News and Views}}</ref><ref>{{cite journal|last=Gerritsma|coauthors=Kirchmair, Zähringer, Solano, Blatt, Roos|title=Quantum simulation of the Dirac equation|journal=Nature|year=2010|url=http://www.nature.com/nature/journal/v463/n7277/full/nature08688.html|volume=463|issue=7277}}</ref>
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| ==Theory==
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| The time-dependent [[Dirac equation]]
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|
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| :<math> H \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi}{\partial t} (\mathbf{x},t) \,\!</math>
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| where <math> H \,\!</math> is the Dirac [[Hamiltonian (quantum mechanics)|Hamiltonian]] for an electron in free space
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| :<math> H = \left(\alpha_0 mc^2 + \sum_{j = 1}^3 \alpha_j p_j \, c\right) \,\!</math>
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| in the [[Heisenberg picture]] implies that any operator Q obeys the equation
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| :<math> -i \hbar \frac{\partial Q}{\partial t} (t)= \left[ H , Q \right] \,\!\;.</math>
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| In particular, the time-dependence of the [[position operator]] is given by
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| :<math> \hbar \frac{\partial x_k}{\partial t} (t)= i\left[ H , x_k \right] = \hbar c\alpha_k \,\!\;</math>
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| where <math>\alpha_k \equiv \gamma_0 \gamma_k</math>.
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| The above equation shows that the operator <math>\alpha_k</math> can be interpreted as the kth component of a "velocity operator". To add time-dependence to <math>\alpha_k</math>, one implements the Heisenberg picture,
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| which says
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| :<math> \alpha_k (t) = e^{i H t / \hbar}\alpha_k e^{-i H t / \hbar}\,\!\;</math>
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| The time-dependence of the velocity operator is given by
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| :<math> \hbar \frac{\partial \alpha_k}{\partial t} (t)= i\left[ H , \alpha_k \right] = 2(i \gamma_k m - \sigma_{kl}p^l) = 2i(p_k-\alpha_kH) \,\!\;</math>
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| where <math>\sigma_{kl} \equiv \frac{i}{2}[\gamma_k,\gamma_l]</math>.
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| Now, because both <math>p_k</math> and <math>H</math> are time-independent, the above equation can easily be integrated twice to
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| find the explicit time-dependence of the position operator. First:
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| :<math>\alpha_k (t) = (\alpha_k (0) - c p_k H^{-1}) e^{-2 i H t / \hbar} + c p_k H^{-1} </math>
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| Then:
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| :<math> x_k(t) = x_k(0) + c^2 p_k H^{-1} t + {1 \over 2 } i \hbar c H^{-1} ( \alpha_k (0) - c p_k H^{-1} ) ( e^{-2 i H t / \hbar } - 1 ) \,\!</math>
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| where <math> x_k(t) \,\!</math> is the position operator at time <math> t \,\!</math>.
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| The resulting expression consists of an initial position, a motion proportional to time, and an unexpected oscillation term with an amplitude equal to the [[Compton wavelength]]. That oscillation term is the so-called "Zitterbewegung".
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| Interestingly, the "Zitterbewegung" term vanishes on taking expectation values for wave-packets that are made up entirely of
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| positive- (or entirely of negative-) energy waves. This can be achieved by taking a [[Foldy-Wouthuysen transformation|Foldy Wouthuysen transformation]]. Thus, we arrive at the interpretation of the "Zitterbewegung" as being caused by
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| interference between positive- and negative-energy wave components.
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| ==See also==
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| * [[Casimir effect]]
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| * [[Lamb shift]]
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| * [[Stochastic electrodynamics]]: Zitterbewegung is explained as an interaction of a classical particle with the [[zero-point field]].
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| ==References and notes==
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| {{reflist}}
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| ==Further reading==
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| * E. Schrödinger, ''Über die kräftefreie Bewegung in der relativistischen Quantenmechanik'' ("On the free movement in relativistic quantum mechanics"), Berliner Ber., pp. 418–428 (1930); Zur Quantendynamik des Elektrons, Berliner Ber, pp. 63–72 (1931)
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| * A. Messiah, ''Quantum Mechanics Volume II'', Chapter XX, Section 37, pp. 950–952 (1962)
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| == External links ==
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| * [http://www.springerlink.com/content/g75q8g1j4h20w5p6/ The Zitterbewegung Interpretation of Quantum Mechanics], an alternative explanation in addition to positive-negative energy states interference.
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| * [http://www.newscientist.com/channel/fundamentals/mg19526112.300-the-word-zitterbewebung.html Zitterbewegung in New Scientist]
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| * [http://geocalc.clas.asu.edu/pdf/ZBWinQM15**.pdf Geometric Algebra in Quantum Mechanics]
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| * [http://physicsworld.com/cws/article/news/41352 Summary of trapped ion simulation]
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| {{Physics-footer}}
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| [[Category:Quantum field theory]]
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| [[Category:German words and phrases]]
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Hello, I'm Meagan, a 26 year old from Lauf, Germany.
My hobbies include (but are not limited to) Gongoozling, Association football and watching How I Met Your Mother.
My web-site: Fifa 15 Coin Generator