We show the effect of quantum-mechanical symmetry on determining the features of two-dimensional few-electron quantum dots, and thereby elucidate the origin of the magic numbers.Recent advances in microfabrication have allowed the creation of quantum dots in semiconductor heterostructures by laterally confining two-dimensional electrons. The confining potential is, to a good approximation, parabolic and a small number ji/ (N=1,2,3, . . . ) of electrons per dot has been achieved experimentally. ' The electronic states of a few-electron system subjected to a strong magnetic field have been studied extensively." For example, to understand the fractional quantum Hall effect, Laughlin first studied the states of a three-electron system in two dimensions in a strong magnetic field and confined by a parabolic potential. Laughlin explicitly constructed the spin-polarized correlated states in the lowest Landau level and showed that they approximated the exact eigenstates well. The ground states turned out to be incompressible since only "magic numbers" of the angular momentum L =3k (/c=1, 2,3, . . . ) of the ground state minimize the Coulomb repulsion. Girvin and Jach extended the analysis to systems containing more electrons. The magic numbers were seen to exist, but the rules explaining them seemed to increase in complexity as the number of particles increased. The role of the electronelectron interaction and the effects of the external magnetic field on few-electron states in quantum dots have been studied by Maksym and Chakraborty (MC). By numerically diagonalizing the Hamiltonian, MC calculated the energy spectra of three-and four-electron quantum dots and pointed out that the angular momentum of the ground state of the elec- ( 4) is for the internal motion, where M=3m*, p&=m*/2, p, =2m*/3, and e2where the term proportional to r, . arises from the confinement. A noteworthy point is that the equivalent particle-
The interaction between two species of atoms in the condensate has been deduced and the trial wave functions that conserve the total spin have been constructed for the states of the ground band. A couple of generalized Gross-Pitaevskii equations have been derived for the spatial wave functions and the energies. The mixture of 23 Na and 87 Rb has been calculated as an example. The density of states is found to exhibit a ladder shape.
The analytical expression of the fractional parentage coefficients for the total spin-states of a spinor N-boson system has been derived. Thereby an S-conserved theory for the spinor BoseEinstein condensation has been proposed. A set of equations has been established to describe the first excited band of the condensates. Numerical solution for 23 Na has been given as an example.PACS numbers: 03.75. Fi, 03.65. Fd In recent years, accompanying the experimental realization of the spinor Bose-Einstein condensation (BEC) [1,2,3,4,5], corresponding theories based on the mean field theory have been developed [6,7,8,9,10,11,12,13]. The total spin S together with its Z-component are actually conserved in the spinor condensates. Furthermore, the interactions are in general spin-dependent. Therefore, the strength of the mean field g that acts on the macroscopically occupied quantum states might depend on S. In previous theories, the S-dependence of the strength g has not yet been perfectly clarified. A primary attempt along this line was proposed in Ref. [14] where the total spin-states have been introduced and the S-dependence of g has been derived analytically.The present paper is a continuation of the Ref. [14]. It is noted that the thermodynamical property of condensates at low temperature depends on their low-lying spectra, this is an interesting topic scarcely touched. The first aim of the paper is to generalize the study of Ref.[14] from the ground band (GB) to excited bands. Rigorous total spin-states with good quantum numbers, S and S Z , and with a specific permutation symmetry (see below) will be introduced, a set of equations is derived to describe, as a first step, the first excited band (FEB) of the spinor BEC.When the total spin states are adopted, how to achieve the matrix elements among them is crucial. The second aim of this paper is to develop a general tool, namely the fractional parentage coefficients, to facilitate the theoretical treatment of spinor N-boson systems. The case of 23 Na as an example will be studied numerically. Before going ahead, we need some knowledge from few-boson systems.Let the interaction between a pair of spin-one bosons beWhere P m and P q are projectors to the two-body spin-states with spin S ij = 0 and 2 (spin 1 is irrelevant since in this case both the spin wavefunction and the spatial wavefunction must be antisymmetric that prevents a point interaction), respectively, g m and g q are the associated strengths, F i is the spin operator of particle i, c 0 = (g m + 2g q )/3 , and c 2 = (g q − g m )/3. Let the N bosons (with mass m) be confined by a harmonic trap with frequency ω. When ω and mω are used as units of energy and length, respectively, the HamiltonianUsing the method as given in [15], the Hamiltonians for N = 3 and 4 two-dimensional systems have been diagonalized. Both the orbital angular momentum L and S are good quantum numbers, therefore the eigenenergies and eigenstates can be denoted as E LS and Ψ LS (there is actually a series of states having the same L an...
The inherent nodal structures of the internal wave functions of two-dimensional few-electron systems have been analyzed to explain the features in the phase diagram and in the filling factors. [S0031-9007(97)04347-0]
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