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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...
, the zitterbewegung ("jittery motion" in German, ) is the predicted rapid oscillatory motion of elementary particles that obey
relativistic wave equations In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
. The existence of such motion was first discussed by
Gregory Breit Gregory Breit (russian: Григорий Альфредович Брейт-Шнайдер, ''Grigory Alfredovich Breit-Shneider''; July 14, 1899, Mykolaiv, Kherson Governorate – September 13, 1981, Salem, Oregon) was a Russian-born Jewish Am ...
in 1928 and later by
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
in 1930 as a result of analysis of the
wave packet In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diff ...
solutions of the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
for relativistic
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s in free space, in which an
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extr ...
between positive and negative
energy state A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The t ...
s 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 In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of , or approximately radians per second. For the
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
, 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 In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
is written as : H \psi (\mathbf,t) = i \hbar \frac (\mathbf,t) , where \hbar is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, \psi(\mathbf,t) is the
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
(
bispinor In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, specifi ...
) of a
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
ic particle
spin-½ In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one ful ...
, and is the Dirac
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
of a
free particle In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. ...
: : H = \beta mc^2 + \sum_^3 \alpha_j p_j c , where m is the mass of the particle, c is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, p_j is the momentum operator, and \beta and \alpha_j are matrices related to the Gamma matrices \gamma_\mu , as \beta=\gamma_0 and \alpha_j=\gamma_0\gamma_j . In the
Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators ( observables and others) incorporate a dependency on time, bu ...
, the time dependence of an arbitrary observable obeys the equation : -i \hbar \frac = \left H , Q \right. In particular, the time-dependence of the position operator is given by : \frac = \frac\left H , x_k \right= c\alpha_k . where is the position operator at time . The above equation shows that the operator can be interpreted as the -th component of a "velocity operator". Note that this implies that : \left\langle \left(\frac\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 , one implements the Heisenberg picture, which says : \alpha_k (t) = e^\frac\alpha_k e^. The time-dependence of the velocity operator is given by : \hbar \frac = i\left H , \alpha_k \right= 2\left(i \gamma_k m - \sigma_p^l\right) = 2i\left(p_k-\alpha_kH\right) , where :\sigma_ \equiv \frac\left gamma_k,\gamma_l\right. Now, because both and are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator. First: :\alpha_k (t) = \left(\alpha_k (0) - c p_k H^\right) e^ + c p_k H^ , and finally : x_k(t) = x_k(0) + c^2 p_k H^ t + \tfrac12 i \hbar c H^ \left( \alpha_k (0) - c p_k H^ \right) \left( e^ - 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 The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It wa ...
. 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 In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED) the negative-energy states are replaced by
positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...
states, and the zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron pairs. 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.


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. 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). Then, in 2013, it was simulated in a setup with
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&nb ...
s. Other proposals for condensed-matter analogues include semiconductor nanostructures,
graphene Graphene () is an allotrope of carbon consisting of a single layer of atoms arranged in a hexagonal lattice nanostructure.
and
topological insulators A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material. A topological insulator is an ...
.


See also

*
Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who predi ...
*
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...


References


Further reading

* {{cite book , first= A. , last= Messiah , title= Quantum Mechanics , volume= II , chapter= XX, Section 37 , pages= 950–952 , date= 1962 , chapter-url= https://archive.org/details/QuantumMechanicsVolumeIi , chapter-format= pdf , isbn= 9780471597681


External links


Zitterbewegung in New Scientist

Geometric Algebra in Quantum Mechanics
Quantum field theory