It might be possible to probe a close analogue of zitterbewegung by recreating the mathematics of Dirac's equation in a quantum wire.
In 1928, British physicist Paul Dirac derived his famous equation for the relativistic quantum mechanics of spin-1/2 particles. Following his unique aesthetic sense for mathematical elegance, Dirac sought a linear differential equation with only a first-order derivative in time, akin to the Schrödinger equation, that would reflect the electron's intrinsic spin and also lead to the relativistic relationship between energy and momentum, E2 = p2c2 + m2c4. Dirac found his equation; and much more.
Dirac's equation has an infinite number of solutions with negative energies. This obviously needs some explaining, as real electrons of positive energy exist stably in free space, and never seem to plunge into that abyss of negativity. Dirac supposed that an invisible sea of negative-energy electrons fills that abyss, preventing ordinary electrons from falling into it. Incredibly, this idea led him to predict the existence of the positron, discovered five years later.
Dirac's equation can no longer be considered mysterious, given its central position in modern physics. But 60 years later, some of the equation's features have yet to be explored experimentally. Modern semiconductor technologies could help to change that, bringing some of the strangest effects finally within experimental reach.
In 1930, Erwin Schrödinger showed that Dirac's equation implies an unexpected relativistic interaction between an electron's translational motion and spin, which should lead to a violent oscillation of the particle at very high frequencies and over distances of roughly one Compton wavelength. This phenomenon of zitterbewegung — from the German word for 'jitter' — has never been observed directly. Given its predicted frequency (about 1021 Hz), it might be some time before a physicist, in the manner of botanist Robert Brown, peers through a microscope of some design to see the electron erratically bouncing around.
But, as several physicists have recently proposed, it might soon be possible to probe a close analogue of zitterbewegung by recreating the mathematics of Dirac's equation in a quantum wire (J. Schliemann et al. Phys. Rev. Lett. 94, 206801; 2005).
For Dirac's electrons, zitterbewegung takes place whenever the electron wavefunction includes both positive and negative energy components; this is generally the case, as it takes both sets of states to build up an arbitrary electronic state. In a semiconducting nanowire, a similar effect takes place for electrons with components in both conduction and valence bands. In some type III–V semiconductors, in particular, the lack of symmetry of the electron hamiltonian under spatial inversion of the atomic lattice induces an interaction between an electron's spin and momentum — as in the Dirac equation.
Calculations, and simulations, suggest that an electron travelling along a narrow quantum wire should undergo a semiconducting version of zitterbewegung — a fluctuating motion in a direction perpendicular to the wire. Its amplitude should grow with the electron's wavelength, and with the steepness of the wire's potential walls. Experimentalists may soon be able to measure this effect.
Quantum-wire zitterbewegung even has a relatavistic origin — linked to very strong electric fields near atomic cores. The effect illustrates, again, the surprising flexibility of semiconductor nanostructures in exploring novel and often exotic phenomena — even some that were predicted 75 years ago.
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Buchanan, M. Dirac gets the jitters. Nature Phys 1, 5 (2005). https://doi.org/10.1038/nphys132
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