In physics, the zitterbewegung (German pronunciation: [ˈtsɪtɐ.bəˌveːɡʊŋ], from German zittern 'to tremble, jitter', and Bewegung 'motion') is the theoretical prediction of a rapid oscillatory motion of elementary particles that obey relativistic wave equations. This prediction 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 an apparent 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.

This apparent oscillatory motion is often interpreted as an artifact of using the Dirac equation in a single particle description and disappears when using quantum field theory. For the hydrogen atom, the zitterbewegung is related to the Darwin term, a small correction of the energy level of the s-orbitals.^[5]

Theory

Free spin-1/2 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 ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle \psi (\mathbf {x} ,t)}$ is the wave function (bispinor) of a fermionic particle spin-1/2, 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 ${\textstyle m}$ is the mass of the particle, ${\textstyle c}$ is the speed of light, ${\textstyle p_{j}}$ is the momentum operator, and ${\displaystyle \beta }$ 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

${\displaystyle -i\hbar {\frac {\partial Q}{\partial t}}=\left.}$

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

${\displaystyle {\frac {\partial x_{k}(t)}{\partial t}}={\frac {i}{\hbar }}\left=c\alpha _{k}}$.

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

The above equation shows that the operator α_k 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 α_k, 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

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

where

${\displaystyle \sigma _{kl}\equiv {\frac {i}{2}}\left.}$

Now, because both p_k 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

${\displaystyle x_k(t)=x_k(0)+c^2{p}_k{H}^{-<!--\; -\; -->1}t+\frac{1}{2}i\mathrm{\hslash <!--\; \hslash \; -->}c{H}^{-<!--\; -\; -->1}({\mathrm{\alpha <!--\; \alpha \; -->}}_k(0)-<!--\; -\; -->c{p}_k{H}^{}}$Zdroj:https://en.wikipedia.org?pojem=Zitterbewegung
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