1,156 research outputs found
Visualizing the collapse and revival of wavepackets in the infinite square well using expectation values
We investigate the short-, medium-, and long-term time dependence of wave
packets in the infinite square well. In addition to emphasizing the appearance
of wave packet revivals, i.e., situations where a spreading wave packet reforms
with close to its initial shape and width, we also examine in detail the
approach to the collapsed phase where the position-space probability density is
almost uniformly spread over the well. We focus on visualizing these phenomena
in both position- and momentum-space as well as by following the time-dependent
expectation values of and uncertainties in position and momentum. We discuss
the time scales for wave packet collapse, using both an autocorrelation
function analysis, as well as focusing on expectation values and find two
relevant time scales which describe different aspects of the decay phase. In an
Appendix, we briefly discuss wave packet revival and collapse in a more
general, one-dimensional power-law potential given by
which interpolates between the case of the harmonic oscillator () and the
infinite well ().Comment: 34 pages, 11 figure
RadiX-Net: Structured Sparse Matrices for Deep Neural Networks
The sizes of deep neural networks (DNNs) are rapidly outgrowing the capacity
of hardware to store and train them. Research over the past few decades has
explored the prospect of sparsifying DNNs before, during, and after training by
pruning edges from the underlying topology. The resulting neural network is
known as a sparse neural network. More recent work has demonstrated the
remarkable result that certain sparse DNNs can train to the same precision as
dense DNNs at lower runtime and storage cost. An intriguing class of these
sparse DNNs is the X-Nets, which are initialized and trained upon a sparse
topology with neither reference to a parent dense DNN nor subsequent pruning.
We present an algorithm that deterministically generates RadiX-Nets: sparse DNN
topologies that, as a whole, are much more diverse than X-Net topologies, while
preserving X-Nets' desired characteristics. We further present a
functional-analytic conjecture based on the longstanding observation that
sparse neural network topologies can attain the same expressive power as dense
counterpartsComment: 7 pages, 8 figures, accepted at IEEE IPDPS 2019 GrAPL workshop. arXiv
admin note: substantial text overlap with arXiv:1809.0524
Quantum mechanical analysis of the equilateral triangle billiard: periodic orbit theory and wave packet revivals
Using the fact that the energy eigenstates of the equilateral triangle
infinite well (or billiard) are available in closed form, we examine the
connections between the energy eigenvalue spectrum and the classical closed
paths in this geometry, using both periodic orbit theory and the short-term
semi-classical behavior of wave packets. We also discuss wave packet revivals
and show that there are exact revivals, for all wave packets, at times given by
where and are the length of one side
and the mass of the point particle respectively. We find additional cases of
exact revivals with shorter revival times for zero-momentum wave packets
initially located at special symmetry points inside the billiard. Finally, we
discuss simple variations on the equilateral
() triangle, such as the half equilateral
() triangle and other `foldings', which have
related energy spectra and revival structures.Comment: 34 pages, 9 embedded .eps figure
Expectation value analysis of wave packet solutions for the quantum bouncer: short-term classical and long-term revival behavior
We discuss the time development of Gaussian wave packet solutions of the
quantum bouncer' (a quantum mechanical particle subject to a uniform downward
force, above an impermeable flat surface). We focus on the evaluation and
visualization of the expectation values and uncertainties of position and
momentum variables during a single quasi-classical period as well as during the
long term collapsed phase and several revivals. This approach complements
existing analytic and numerical analyses of this system, as well as being
useful for comparison with similar results for the harmonic oscillator and
infinite well cases.Comment: 20 pages, 7 separate .ps figure
- …