1,406 research outputs found
On separable Schr\"odinger equations
We classify (1+3)-dimensional Schr\"odinger equations for a particle
interacting with the electromagnetic field that are solvable by the method of
separation of variables. As a result, we get eleven classes of the
electromagnetic vector potentials of the electromagnetic field , providing separability of the
corresponding Schr\"odinger equations. It is established, in particular, that
the necessary condition for the Schr\"odinger equation to be separable is that
the magnetic field must be independent of the spatial variables. Next, we prove
that any Schr\"odinger equation admitting variable separation into second-order
ordinary differential equations can be reduced to one of the eleven separable
Schr\"odinger equations mentioned above and carry out variable separation in
the latter. Furthermore, we apply the results obtained for separating variables
in the Hamilton-Jacobi equation.Comment: 30 pages, LaTe
Coupling of non-crossing wave modes in a two-dimensional plasma crystal
We report an experimental observation of coupling of the transverse vertical
and longitudinal in-plane dust-lattice wave modes in a two-dimensional complex
plasma crystal in the absence of mode crossing. A new large diameter rf plasma
chamber was used to suspend the plasma crystal. The observations are confirmed
with molecular-dynamics simulations. The coupling manifests itself in traces of
the transverse vertical mode appearing in the measured longitudinal spectra and
vice versa. We calculate the expected ratio of the trace to the principal mode
with a theoretical analysis of the modes in a crystal with finite temperature
and find good agreement with the experiment and simulations.Comment: 4 figures, 5 pages, accepted for publication in PRL Nov 201
Observation of particle pairing in a two-dimensional plasma crystal
The observation is presented of naturally occurring pairing of particles and
their cooperative drift in a two-dimensional plasma crystal. A single layer of
plastic microspheres was suspended in the plasma sheath of a capacitively
coupled rf discharge in argon at a low pressure of 1 Pa. The particle dynamics
were studied by combining the top-view and side-view imaging of the suspension.
Cross analysis of the particle trajectories allowed us to identify naturally
occurring metastable pairs of particles. The lifetime of pairs was long enough
for their reliable identification.Comment: 5 pages, 4 figure
First direct measurement of optical phonons in 2D plasma crystals
Spectra of phonons with out-of-plane polarization were studied experimentally
in a 2D plasma crystal. The dispersion relation was directly measured for the
first time using a novel method of particle imaging. The out-of-plane mode was
proven to have negative optical dispersion, comparison with theory showed good
agreement. The effect of the plasma wakes on the dispersion relation is briefly
discussed.Comment: submitted to Physical Review Letter
Synchronization of particle motion in compressed two-dimensional plasma crystals
The collective motion of dust particles during the mode-coupling induced
melting of a two-dimensional plasma crystal is explored in molecular dynamics
simulations. The crystal is compressed horizontally by an anisotropic
confinement. This compression leads to an asymmetric triggering of the
mode-coupling instability which is accompanied by alternating chains of
in-phase and anti-phase oscillating particles. A new order parameter is
proposed to quantify the synchronization with respect to different directions
of the crystal. Depending on the orientation of the confinement anisotropy,
mode-coupling instability and synchronized motion are observed in one or two
directions. Notably, the synchronization is found to be direction-dependent.
The good agreement with experiments suggests that the confinement anisotropy
can be used to explain the observed synchronization process.Comment: 6 pages, 4 figure
Wave mode coupling due to plasma wakes in two-dimensional plasma crystals: In-depth view
Experiments with two-dimensional (2D) plasma crystals are usually carried out
in rf plasma sheaths, where the interparticle interactions are modified due to
the presence of plasma wakes. The wake-mediated interactions result in the
coupling between wave modes in 2D crystals, which can trigger the mode-coupling
instability and cause melting. The theory predicts a number of distinct
fingerprints to be observed upon the instability onset, such as the emergence
of a new hybrid mode, a critical angular dependence, a mixed polarization, and
distinct thresholds. In this paper we summarize these key features and provide
their detailed discussion, analyze the critical dependence on experimental
parameters, and highlight the outstanding issues
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