This causes to coincide with the eigenfunctions of the Solution is scarred on precisely these orbits (d). A semi-classical simulation (c) shows why this is so. Repressing the classical chaos, through probabilities clumping on the repelling Shows ‘scarring’ of the wave function along these repelling orbits, thus The quantum solution of the stadium potential well The stadium is densely filled with repelling periodic orbits, three of whichĪre shown in black in (d). Particularly complete expose of the methods involved is provided in their The convergence of the trace formula and its modifications remain unanswered. The authors comment: “Accurate excited-state eigenvalues have been computedįrom knowledge of relatively few periodic orbits. Trajectories connecting a given location to another in the region.Īmong others have used the semi-classical orbit-based approach to develop the Semi-classical approach uses the least repelling periodic orbits in theĬlassical stadium to generate amplitudes based on the path lengths of all classical Propagation approach is discretely modeled in the CA below. In the quantumĪpproach a wave packet is propagated throughout the potential well using a timeĭependent Schrödinger equation via Fourier transform techniques. Have been used to model systems such as the stadium billiard. The general similarity of the trace formula to the Riemann zetaĪnd this function’s GUE type statistics shared by quantum chaotic systems (figĢe) has led to extensive attempts to solve quantum chaotic systems through theīoth quantum and semi-classical approaches On reordering the sum terms), the eigenvalues will appear as singularities in Sum converges (this is generally possible only with some difficulty and depends From the left hand side of the equation it can be seen that if the The stability matrix records the sensitivity of the orbit to changes in initialĬonditions. The trace formula for a particle of mass m and momentum inside a box with arbitrarily shaped walls isįunction giving the mean density of states, summed over all orbits of length with the phase shiftĬounting the focal points and twice the number of reflections off the walls and Green’s function of wave propagation over all coordinates so that it can beĮxpressed in terms of the repelling orbits hidden within the classically In 1970 ĭeveloped a semi-classical method using a trace formula, which integrates the Understanding these quantum solutions so their dynamics can be more directlyĬhaotic wave function, given its boundary constraints proved illusive for theįounders of quantum theory in the case of the helium nucleus. ![]() Part of the purpose of this paper is to simplify (the classical Coliseum stadium shape consisting of a rectangle or squareĬapped off with semi-circular discs) proved to be initially intractable andĮven in the post-modern era of revived semi-classical methods computationallyĬomplex to the point of being counter-intuitive. Quicktime movie of the wave function CA operatingĮstimation of the quantum wave functions inĬhaotic systems, from the nucleus of atoms from helium to uranium to theĮigenfunctions of a wave-particle in the Bunimovich stadium The probability distributions give better resolution of the scarring. Phase amplitudes and lower two the corresponding superimposed probabilityĭistributions of the squared amplitudes, showing evident scarring. Finally a simplified semi-classical algorithm is developed to show the comparison between this and the quantum wave function method.įig 1: Evidence of scarring: Superposition of the first 3300 iterations of a wave function initiated by two different generating functions (a and c in fig 4) with wavelength chosen to be a fraction of the associated classical repelling orbit. The classical orbits are computed by solving the reflection equations at the classical boundary thus giving direct insights into the wave functions and eigenstates of the quantum stadium. The quantum wave functions are modeled using a cellular automaton (CA) simulating a Hamiltonian wave function with discrete (square pixel) boundary conditions approaching the stadium in the classical limit. The simulations use three complementary methods. You can now run the wave function CAs in my open source application CA2D for Mac!Ībstract: This paper explores quantum and classical chaos in the stadium billiard using Matlab simulations to investigate the behavior of wave functions in the stadium and the corresponding classical orbits believed to underlie wave function scarring. ![]() Mathematics Department – University of Auckland – Exploring Quantum, Classical and Semiclassical
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