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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-

Multienzyme cascade approaches for the synthesis of optically pure molecules from simple achiral compounds are desired. Herein, a cofactor self‐sufficient cascade protocol for the asymmetric amination of racemic secondary alcohols to the corresponding chiral amines was successfully constructed by employing an alcohol dehydrogenase and a newly developed amine dehydrogenase. The compatibility and the identical cofactor dependence of the two enzymes led to an ingenious in situ cofactor recycling system in the one‐pot synthesis. The artificial redox‐neutral cascade process allowed the transformation of racemic secondary alcohols into enantiopure amines with considerable conversions (up to 94 %) and >99 % enantiomeric excess at the expense of only ammonia; this method thus represents a concise and efficient route for the asymmetric synthesis of chiral amines.

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