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WOLFRAM|DEMONSTRATIONS PROJECT

Relativistic Quantum Dynamics in 1D and the Klein Paradox

​
run
hold state
mode
Dirac
Schrödinger
both
Set the initial state ψ.Restarts if running.
n
points
122
σ
0.2
x
0.6
p /
p
max
0.05
Set the Hamiltonian H.
Readjusts the time step if running.
m /
m
max
0.2
κ
8
λ
30.
λ
bell
1.
σ
bell
0.14
x
bell
0.
Change speed and appearance.Does not restart if running.
dt
0.
y
U
1.1
-
y
L
0.5
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.
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