Relativistic Quantum Dynamics in 1D and the Klein Paradox

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This Demonstration shows the dynamics of a wave packet as determined by Dirac's relativistic equation and a stationary electric potential. For comparison the same physical situation is also treated with Schrödinger's nonrelativistic equation. The solution method is discretization on a 1D periodic lattice together with a modified leapfrog integrator. There are several control options that are explained by tooltips. Most importantly, you can set the initial state, an electric potential, and start the simulation by activating the "run" checkbox. It is shown visually that the mass of the relativistic particle contributes to its energy and thus leads to a high-frequency phase rotation that is missing in the dynamics of the Schrödinger wave function.

Contributed by: Ulrich Mutze (April 2011)
Open content licensed under CC BY-NC-SA



The Dirac equation governs the time-dependence of a complex-valued four-component field . We consider only fields that do not depend on and , which reduces to the equation , , and a corresponding pair of equations for and . Here units are used in which ; is the mass of the Dirac particle and an electric potential that does not depend on , , or . As a Dirac wavefunction in 1D we thus take , with the Schrödinger analog having .

The snapshots deal with Klein's paradox, which is described in most textbooks covering the Dirac equation, for example [1]. As noticed earlier [3], wave packets hitting a potential barrier show no paradoxical behavior in situations where monochromatic plane waves show an anomalously high amplitude of the reflected wave. At present there seems to be no solid understanding for the different behavior of wave packets and monochromatic waves in this case. This is not too surprising since in nonrelativistic scattering theory it is already a major enterprise to learn how to properly relate these two pictures (stationary versus time-dependent scattering theory).

Snapshot 1: This initial state shows a Gaussian wave packet with momentum directed toward the central potential wall. The only nonvanishing spinor component is . This choice is enforced since in the Schrödinger case, which I want to treat on an equal footing, there is no analog to the component (shown as cyan and magenta curves).

Snapshot 2: The evolution creates the component of the wavefunction. This new component has a large subcomponent of negative energy that lets it move in the opposite direction (i.e. to the left). As already observed in [3], the packet-splitting would be avoided if one would define the initial state with the correct admixture of . Anyway, the part of the split packet still moving to the right is just the packet that such an elaborate state preparation would give in the first place. Notice that the causal structure of our dynamical system ensures that the two packets can influence each other only if they overlap. The Schrödinger packet just runs into the wall and is significantly deformed.

Snapshot 3: Due to momentum conservation, the ejection of the pure packet slows down the main packet, which hits the wall only then. The Schrödinger packet just completes the reflection phase.

Snapshot 4: The reflection occurs for both packets. Form and size are the same as before the collision. In the Dirac case this refers to the wave packet after it has ejected the pure packet.

Snapshot 5: No collision occurs.

Snapshot 6–7: Collisions are as expected.

The autorun feature here simply illustrates the selection options for the initial state and the potential. Running the simulation with different parameters in autorun mode seems difficult to implement.


[1] C. Itzykson and J.–B. Zuber, Quantum Field Theory, New York: McGraw–Hill, 1980.

[2] I. Bialynicki–Birula, "Weyl, Dirac, and Maxwell Equations on a Lattice as Unitary Cellular Automata," Physical Review D, 49(12), 1994 pp. 6920–6927.

[3] N. Simicevic, "Finite Difference-Time Domain Solution of Dirac Equation and the Klein Paradox," arXiv:0901.3765v1 [quant-ph] (2009).

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