208 research outputs found
Distortion of Wigner molecules : pair function approach
We considered a two dimensional three electron quantum dot in a magnetic
field in the Wigner limit. A unitary coordinate transformation decouples the
Hamiltonian (with Coulomb interaction between the electrons included) into a
sum of three independent pair Hamiltonians. The eigen-solutions of the pair
Hamiltonian provide a spectrum of pair states. Each pair state defines the
distance of the two electrons involved in this state. In the ground state for
given pair angular momentum , this distance increases with increasing .
The pair states have to be occupied under consideration of the Pauli exclusion
principle, which differs from that for one-electron states and depends on the
total spin and the total orbital angular momentum (sum over
all pair angular momenta). We have shown that the three electrons in the ground
state of the Wigner molecule form an equilateral triangle (as might be
expected) only, if the state is a quartet () and the orbital angular
momentum is a magic quantum number ( integer). Otherwise the
triangle in the ground state is isosceles. For one of the sides is
longer and for one of the sides is shorter than the other two
Analytic Solution of a Relativistic Two-dimensional Hydrogen-like Atom in a Constant Magnetic Field
We obtain exact solutions of the Klein-Gordon and Pauli Schroedinger
equations for a two-dimensional hydrogen-like atom in the presence of a
constant magnetic field. Analytic solutions for the energy spectrum are
obtained for particular values of the magnetic field strength. The results are
compared to those obtained in the non-relativistic and spinless case. We obtain
that the relativistic spectrum does not present s states.Comment: RevTeX, 8 pages, to be published in Phys. Lett.
Violation of non-interacting -representability of the exact solutions of the Schr\"odinger equation for a two-electron quantum dot in a homogeneous magnetic field
We have shown by using the exact solutions for the two-electron system in a
parabolic confinement and a homogeneous magnetic field [ M.Taut, J Phys.A{\bf
27}, 1045 (1994) ] that both exact densities (charge- and the paramagnetic
current density) can be non-interacting -representable (NIVR) only in a
few special cases, or equivalently, that an exact Kohn-Sham (KS) system does
not always exist. All those states at non-zero can be NIVR, which are
continuously connected to the singlet or triplet ground states at B=0. In more
detail, for singlets (total orbital angular momentum is even) both
densities can be NIVR if the vorticity of the exact solution vanishes. For
this is trivially guaranteed because the paramagnetic current density
vanishes. The vorticity based on the exact solutions for the higher
does not vanish, in particular for small r. In the limit this can
even be shown analytically. For triplets ( is odd) and if we assume
circular symmetry for the KS system (the same symmetry as the real system) then
only the exact states with can be NIVR with KS states having angular
momenta and . Without specification of the symmetry of the KS
system the condition for NIVR is that the small-r-exponents of the KS states
are 0 and 1.Comment: 18 pages, 4 figure
Charged particles in external fields as physical examples of quasi-exactly solvable models: a unified treatment
We present a unified treatment of three cases of quasi-exactly solvable
problems, namely, charged particle moving in Coulomb and magnetic fields, for
both the Schr\"odinger and the Klein-Gordon case, and the relative motion of
two charged particles in an external oscillator potential. We show that all
these cases are reducible to the same basic equation, which is quasi-exactly
solvable owing to the existence of a hidden algebraic structure. A
systematic and unified algebraic solution to the basic equation using the
method of factorization is given. Analytic expressions of the energies and the
allowed frequencies for the three cases are given in terms of the roots of one
and the same set of Bethe ansatz equations.Comment: RevTex, 15 pages, no figure
Wigner Crystallization of a two dimensional electron gas in a magnetic field: single electrons versus electron pairs at the lattice sites
The ground state energy and the lowest excitations of a two dimensional
Wigner crystal in a perpendicular magnetic field with one and two electrons per
cell is investigated. In case of two electrons per lattice site, the
interaction of the electrons {\em within} each cell is taken into account
exactly (including exchange and correlation effects), and the interaction {\em
between} the cells is in second order (dipole) van der Waals approximation. No
further approximations are made, in particular Landau level mixing and {\em
in}complete spin polarization are accounted for. Therefore, our calculation
comprises a, roughly speaking, complementary description of the bubble phase
(in the special case of one and two electrons per bubble), which was proposed
by Koulakov, Fogler and Shklovskii on the basis of a Hartree Fock calculation.
The phase diagram shows that in GaAs the paired phase is energetically more
favorable than the single electron phase for, roughly speaking, filling factor
larger than 0.3 and density parameter smaller than 19 effective Bohr
radii (for a more precise statement see Fig.s 4 and 5). If we start within the
paired phase and increase magnetic field or decrease density, the pairs first
undergo some singlet- triplet transitions before they break.Comment: 11 pages, 7 figure
Solution of the Schr\"odinger Equation for Quantum Dot Lattices with Coulomb Interaction between the Dots
The Schr\"odinger equation for quantum dot lattices with non-cubic,
non-Bravais lattices built up from elliptical dots is investigated. The Coulomb
interaction between the dots is considered in dipole approximation. Then only
the center of mass (c.m.) coordinates of different dots couple with each other.
This c.m. subsystem can be solved exactly and provides magneto- phonon like
collective excitations. The inter-dot interaction is involved only through a
single interaction parameter. The relative coordinates of individual dots form
decoupled subsystems giving rise to intra-dot excitations. As an example, the
latter are calculated exactly for two-electron dots.
Emphasis is layed on qualitative effects like: i) Influence of the magnetic
field on the lattice instability due to inter-dot interaction, ii) Closing of
the gap between the lower and the upper c.m. mode at B=0 for elliptical dots
due to dot interaction, and iii) Kinks in the single dot excitation energies
(versus magnetic field) due to change of ground state angular momentum. It is
shown that for obtaining striking qualitative effects one should go beyond
simple cubic lattices with spherical dots. We also prove a more general version
of the Kohn Theorem for quantum dot lattices. It is shown that for observing
effects of electron- electron interaction between the dots in FIR spectra
(breaking Kohn's Theorem) one has to consider dot lattices with at least two
dot species with different confinement tensors.Comment: 11 figures included as ps-file
A new quasi-exactly solvable problem and its connection with an anharmonic oscillator
The two-dimensional hydrogen with a linear potential in a magnetic field is
solved by two different methods. Furthermore the connection between the model
and an anharmonic oscillator had been investigated by methods of KS
transformation
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