Quantum scar

In physics, and especially quantum chaos, a wavefunction scar is an enhancement (i.e. increased norm squared) of an eigenfunction along unstable classical periodic orbits. Scars are related to the correspondence principle. The existence of the scar is directly implied by the Ehrenfest theorem. While the only way the wave function of the stationary system can propagate is through the evolution of quantum phases of its components and the corresponding Gaussian wave packet must on the other hand move along the classical stable or unstable trajectory there must be therefore eigenstates along the periodic trajectory so large that those components lead to this persistent propagation. Quantum scars in quantum stadium (with the wave function vanishing on the stadium shape) can be readily understood as periodic solutions of the Klein–Gordon equation with the imaginary time (or the propagation velocity) when the along width coordinate of the stadium is the time and the scar energy is the square of the relativistic mass energy at rest. The Dirichlet problem is then equivalent to the pseudo-relativistic particle in the infinite potential well when the walls are moving in time according as two semi-circle functions and they are one-dimensional wave packets propagating in time that look scars as quantum carpets. At the first approximation the standing wave adiabatic solutions of the Klein–Gordon equation with slowly varying frequencies on the line with the variable length (the parallel distance between the two semi-circles) can be taken as the basis to span the eigenstates of the stadium. The scars are then the effect of the wave beat between those various oscillatory components that leads to the probability amplification along the classical periodic trajectory. The reflections of scars from parallel stadium edges are then the volumetric Klein paradox (particle probability coming from nowhere at the whole volume or from all points of the one-dimensional space due to the imaginary propagation velocity).