Abstract
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. The simulations use three complementary methods. The quantum wave functions are modeled using a cellular automaton simulating a Hamiltonian wave function with discrete (square pixel) boundary conditions approaching the stadium in the classical limit. 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. Finally, a simplified semi-classical algorithm is developed to show the comparison between this and the quantum wave function method. Quanta 2014; 3: 16–31.
Highlights
2 The Quantum Cellular Automaton integrating their effect: To get a more direct picture of the process of wave spreading and wave function scarring [see panels (a-d) in Figure 2], this simulation depends on a simple wave propagation formula for a cellular automaton which preserves the 1 mi (Fi(t vi(t + ∆t) = vi(t) + ai(t)∆t ψi(t + ∆t) = ψi(t) + vi(t + ∆t)∆t essential nature of Hamiltonian dynamics and the Green’s The simulation uses a variety of generating functions to function governing transmission of wave amplitudes from set up excitations whose period corresponds to a fraction point to point
The dynamic can be described in terms of the cillating wave propagator to the member of the form [7]: Markov partition of phase space determined by the orbits which strike or emanate from the four junctions at (±1, ±1), labelled in red, magenta, green and blue dots in panels (d-g) of Figure 9
To explore a simplified semi-classical process, illustrat- 6 Summary ing some of the principles of the above approach, we developed a Matlab program superimposing successive The central method introduced in the paper, provides reflections of a 360 degree fan of 212 − 218 rays from a straightforward cellular automata model for a semithe centre of the stadium with an oscillating propagator classical approach to quantum chaos, more natural than having period a fraction of the linear dimensions of the the semi-classical methods of [5,6,7,8], demonstrating quanstadium
Summary
The authors to solve quantum chaotic systems through the paradigm comment: of the zeta zeros (for a detailed discussion on the quantum chaos connection with the Riemann zeta function see Accurate excited-state eigenvalues have been section 7). Both quantum and semi-classical approaches have been used to model systems such as the stadium billiard. This wave is provided in their Physical Review E paper [7]
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