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. The word was first applied to the relativistic motion of free electrons by Erwin Schrödinger in 1930 in his analysis of wave packet solutions of the Dirac equation for relativistic electrons in free space. These exhibit interference between positive and negative energy states, which 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, which is twice the Compton angular frequency. The oscillatory Zitterbewegung motion is often interpreted as an artifact of using the Dirac equation in a single particle description and disappears in 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.
Theory
Free spin-1/2 fermion The time-dependent Dirac equation is written as
H ψ ( x , t ) = i ℏ ∂ ψ ∂ t ( x , t ) {\displaystyle H\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {x} ,t)} , where ℏ {\displaystyle \hbar } is the reduced Planck constant, ψ ( x , t ) {\displaystyle \psi (\mathbf {x} ,t)} is the wave function (Dirac spinor) of a fermionic particle spin-1/2, and H is the Dirac Hamiltonian of a free particle:
H = β m c 2 + ∑ j = 1 3 α j p j c {\displaystyle H=\beta mc^{2}+\sum _{j=1}^{3}\alpha _{j}p_{j}c} , where m {\textstyle m} is the mass of the particle, c {\textstyle c} is the speed of light, p j {\textstyle p_{j}} is the momentum operator, and β {\displaystyle \beta } and α j {\displaystyle \alpha _{j}} are matrices related to the Gamma matrices γ μ {\textstyle \gamma _{\mu }} , as β = γ 0 {\textstyle \beta =\gamma _{0}} and α j = γ 0 γ j {\textstyle \alpha _{j}=\gamma _{0}\gamma _{j}} . In the Heisenberg picture, the time dependence of an arbitrary observable Q obeys the equation
− i ℏ d Q d t = [ H , Q ] . {\displaystyle -i\hbar {\frac {dQ}{dt}}=\left[H,Q\right].}
In particular, the time-dependence of the position operator is given by
d x k ( t ) d t = i ℏ [ H , x k ] = c α k {\displaystyle {\frac {dx_{k}(t)}{dt}}={\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 α k {\displaystyle \alpha _{k}} can be interpreted as the k-th component of a "velocity operator". Note that this implies that
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