Zitterbewegung

In physics, the zitterbewegung ("jittery motion" in German, German pronunciation: [ˈtsɪtɐ.bəˌveːɡʊŋ]) is the predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first discussed by Gregory Breit in 1928[1][2] and later by Erwin Schrödinger in 1930[3][4] as a result of analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (up to the speed of light) of the position of an electron around the median, with an angular frequency of 2mc2/ℏ, or approximately 1.6×1021 radians per second. For the hydrogen atom, zitterbewegung can be invoked as a heuristic way to derive the Darwin term, a small correction of the energy level of the s-orbitals.
Theory
Free fermion

The time-dependent Dirac equation is written as

\( {\displaystyle H\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {x} ,t)}, \)

where \( \hbar \( is the reduced Planck constant, \( {\displaystyle \psi (\mathbf {x} ,t)} \) is the wave function (bispinor) of a fermionic particle spin-½, and H is the Dirac Hamiltonian of a free particle:

\( {\displaystyle H=\beta mc^{2}+\sum _{j=1}^{3}\alpha _{j}p_{j}c}, \)

where m is the mass of the particle, c is the speed of light, \( {\textstyle p_{j}} \) is the momentum operator, and \( \beta \) and \( {\displaystyle \alpha _{j}} \) are matrices related to the Gamma matrices \( {\textstyle \gamma _{\mu }} \), as \( {\textstyle \beta =\gamma _{0}} \) and \( {\textstyle \alpha _{j}=\gamma _{0}\gamma _{j}} \).

In the Heisenberg picture, the time dependence of an arbitrary observable Q obeys the equation

\( . {\displaystyle -i\hbar {\frac {\partial Q}{\partial t}}=\left[H,Q\right].} \)

In particular, the time-dependence of the position operator is given by

\( {\displaystyle {\frac {\partial x_{k}(t)}{\partial t}}={\frac {i}{\hbar }}\left[H,x_{k}\right]=c\alpha _{k}}. \)

where xk(t) is the position operator at time t.

The above equation shows that the operator αk can be interpreted as the k-th component of a "velocity operator".

Note that this implies that

\( {\displaystyle \left\langle \left({\frac {\partial x_{k}(t)}{\partial t}}\right)^{2}\right\rangle =c^{2}}, \)

as if the "root mean square speed" in every direction of space is the speed of light.

To add time-dependence to αk, one implements the Heisenberg picture, which says

\( {\displaystyle \alpha _{k}(t)=e^{\frac {iHt}{\hbar }}\alpha _{k}e^{-{\frac {iHt}{\hbar }}}}. \)

The time-dependence of the velocity operator is given by

\( {\displaystyle \hbar {\frac {\partial \alpha _{k}(t)}{\partial t}}=i\left[H,\alpha _{k}\right]=2\left(i\gamma _{k}m-\sigma _{kl}p^{l}\right)=2i\left(p_{k}-\alpha _{k}H\right)}, \)

where

\( {\displaystyle \sigma _{kl}\equiv {\frac {i}{2}}\left[\gamma _{k},\gamma _{l}\right].} \)

Now, because both pk and H are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator.

First:

\( {\displaystyle \alpha _{k}(t)=\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)e^{-{\frac {2iHt}{\hbar }}}+cp_{k}H^{-1}}, \)

and finally

\( {\displaystyle x_{k}(t)=x_{k}(0)+c^{2}p_{k}H^{-1}t+{\tfrac {1}{2}}i\hbar cH^{-1}\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)\left(e^{-{\frac {2iHt}{\hbar }}}-1\right)}. \)

The resulting expression consists of an initial position, a motion proportional to time, and an oscillation term with an amplitude equal to the reduced Compton wavelength. That oscillation term is the so-called zitterbewegung.
Interpretation as an artifact

In quantum mechanics, the zitterbewegung term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves. The standard relativistic velocity can be recovered by taking a Foldy–Wouthuysen transformation, when the positive and negative components are decoupled. Thus, we arrive at the interpretation of the zitterbewegung as being caused by interference between positive- and negative-energy wave components.

In quantum electrodynamics (QED) the negative-energy states are replaced by positron states, and the zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron pairs.[5]

More recently, it has been noted that in the case of free particles it could just be an artifact of the simplified theory. Zitterbewegung appear as due to the "small components" of the Dirac 4-spinor, due to a little bit of antiparticle mixed up in the particle wavefunction for a nonrelativistic motion. It doesn't appear in the correct second quantized theory, or rather, it is resolved by using Feynman propagators and doing QED. Nevertheless, it is an interesting way to understand certain QED effects heuristically from the single particle picture. [6]
Experimental simulation

Zitterbewegung of a free relativistic particle has never been observed directly, although some authors believe they have found evidence in favor of its existence.[7] It has also been simulated twice in model systems that provide condensed-matter analogues of the relativistic phenomenon. The first example, in 2010, placed a trapped ion in an environment such that the non-relativistic Schrödinger equation for the ion had the same mathematical form as the Dirac equation (although the physical situation is different).[8][9] Then, in 2013, it was simulated in a setup with Bose–Einstein condensates.[10]

Other proposals for condensed-matter analogues include semiconductor nanostructures, graphene and topological insulators.[11][12][13][14]
See also

Casimir effect
Lamb shift

References

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Greiner, Walter (1995). Relativistic Quantum Mechanics. doi:10.1007/978-3-642-88082-7. ISBN 978-3-540-99535-7. S2CID 124404090.
Schrödinger, E. (1930). Über die kräftefreie Bewegung in der relativistischen Quantenmechanik [On the free movement in relativistic quantum mechanics] (in German). pp. 418–428. OCLC 881393652.
Schrödinger, E. (1931). Zur Quantendynamik des Elektrons [Quantum Dynamics of the Electron] (in German). pp. 63–72.
Zhi-Yong, W., & Cai-Dong, X. (2008). Zitterbewegung in quantum field theory. Chinese Physics B, 17(11), 4170.
"Dirac equation - is Zitterbewegung an artefact of single-particle theory?".
Catillon, P.; Cue, N.; Gaillard, M. J.; et al. (2008-07-01). "A Search for the de Broglie Particle Internal Clock by Means of Electron Channeling". Foundations of Physics. 38 (7): 659–664. doi:10.1007/s10701-008-9225-1. ISSN 1572-9516. S2CID 121875694.
Wunderlich, Christof (2010). "Quantum physics: Trapped ion set to quiver". Nature News and Views. 463 (7277): 37–39. doi:10.1038/463037a. PMID 20054385.
Gerritsma; Kirchmair; Zähringer; Solano; Blatt; Roos (2010). "Quantum simulation of the Dirac equation". Nature. 463 (7277): 68–71. arXiv:0909.0674. Bibcode:2010Natur.463...68G. doi:10.1038/nature08688. PMID 20054392. S2CID 4322378.
Leblanc; Beeler; Jimenez-Garcia; Perry; Sugawa; Williams; Spielman (2013). "Direct observation of zitterbewegung in a Bose–Einstein condensate". New Journal of Physics. 15 (7): 073011. arXiv:1303.0914. doi:10.1088/1367-2630/15/7/073011. S2CID 119190847.
Schliemann, John (2005). "Zitterbewegung of Electronic Wave Packets in III-V Zinc-Blende Semiconductor Quantum Wells". Physical Review Letters. 94 (20): 206801. arXiv:cond-mat/0410321. doi:10.1103/PhysRevLett.94.206801. PMID 16090266. S2CID 118979437.
Katsnelson, M. I. (2006). "Zitterbewegung, chirality, and minimal conductivity in graphene". The European Physical Journal B. 51 (2): 157–160. arXiv:cond-mat/0512337. doi:10.1140/epjb/e2006-00203-1. S2CID 119353065.
Dóra, Balász; Cayssol, Jérôme; Simon, Ference; Moessner, Roderich (2012). "Optically engineering the topological properties of a spin Hall insulator". Physical Review Letters. 108 (5): 056602. arXiv:1105.5963. doi:10.1103/PhysRevLett.108.056602. PMID 22400947. S2CID 15507388.

Shi, Likun; Zhang, Shoucheng; Cheng, Kai (2013). "Anomalous Electron Trajectory in Topological Insulators". Physical Review B. 87 (16). arXiv:1109.4771. doi:10.1103/PhysRevB.87.161115. S2CID 118446413.