About the quantum Talbot effect on the sphere
- URL: http://arxiv.org/abs/2302.11063v2
- Date: Thu, 27 Apr 2023 00:48:38 GMT
- Title: About the quantum Talbot effect on the sphere
- Authors: Fernando Chamizo and Osvaldo Santillan
- Abstract summary: The Schr"odinger equation on a circle with an initially localized profile of the wave function is known to give rise to revivals or replications.
The structure of singularities of the resulting wave function is characterized in detail.
It is suggested that, differently from the circle case, these regions are not lines but instead some specific set of points along the sphere.
- Score: 77.34726150561087
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: The Schr\"odinger equation on a circle with an initially localized profile of
the wave function is known to give rise to revivals or replications, where the
probability density of the particle is partially reproduced at rational times.
As a consequence of the convolutional form of the general solution it is
deduced that a piecewise constant initial wave function remains piecewise
constant at rational times as well. For a sphere instead, it is known that this
piecewise revival does not necessarily occur, indeed the wave function becomes
singular at some specific locations at rational times. It may be desirable to
study the same problem, but with an initial condition being a localized Dirac
delta instead of a piecewise constant function, and this is the purpose of the
present work. By use of certain summation formulas for the Legendre polynomials
together with properties of Gaussian sums, it is found that revivals on the
sphere occur at rational times for some specific locations, and the structure
of singularities of the resulting wave function is characterized in detail. In
addition, a partial study of the regions where the density vanishes, named
before valley of shadows in the context of the circle, is initiated here. It is
suggested that, differently from the circle case, these regions are not lines
but instead some specific set of points along the sphere. A conjecture about
the precise form of this set is stated and the intuition behind it is
clarified.
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