Zitterbewegung

Zitterbewegung ('trembling motion' in German) is a predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. 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 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. A reexamination of Dirac theory, however, shows that interference between positive and negative energy states may not be a necessary criterion for observing zitterbewegung. Zitterbewegung ('trembling motion' in German) is a predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. 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 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. A reexamination of Dirac theory, however, shows that interference between positive and negative energy states may not be a necessary criterion for observing zitterbewegung. For the hydrogen atom, the zitterbewegung produces the Darwin term which plays the role in the fine structure as a small correction of the energy level of the s-orbitals. The time-dependent Dirac equation is written as where ℏ {displaystyle hbar } is the (reduced) Planck constant, ψ ( x , t ) {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: where m { extstyle m} is the mass of the particle, c { extstyle c} is the speed of light, p j { extstyle p_{j}} is the momentum operator, and β {displaystyle eta } and α j {displaystyle alpha _{j}} are matrices related to the Gamma matrices γ μ { extstyle gamma _{mu }} , as β = γ 0 { extstyle eta =gamma _{0}} and α j = γ 0 γ j { extstyle alpha _{j}=gamma _{0}gamma _{j}} . The Heisenberg picture implies that any operator Q obeys the equation In particular, the time-dependence of the position operator is given by 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'. To add time-dependence to αk, one implements the Heisenberg picture, which says