Question

kinetic energy is 2d or 3d

Answer 1

Kinetic related ground state properties of a two-electron 2D quantum dot in a magnetic field and a 3D quantum dot (Hooke’s atom) are compared in the Wigner high (HEC) and low (LEC) electron correlation regimes. The HEC regime corresponds to low densities sufficient for the creation of a Wigner molecule. The LEC regime densities are similar to those of natural atoms and molecules. The results are determined employing exact closed-form analytical solutions of the Schrödinger-Pauli and Schrödinger equations, respectively. The properties studied are the local and nonlocal quantal sources of the density and the single particle density matrix; the kinetic energy density; the kinetic ‘force’ and its divergence; the kinetic field; and the kinetic energy. The correlation-kinetic energy is obtained by mapping the 2D and 3D quantum dots via quantal density functional theory to systems of noninteracting fermions possessing the same density and physical current density for the former, and the same density for the latter. A key observation is that the structure of the 2D and 3D system properties within a specific electron correlation regime are similar. The quantal compression of the kinetic energy density about the center of the quantum dots in the HEC regime, and the quantal decompression away from the center in the LEC regime is explained via the structure of the kinetic ‘force’ and of its divergence, as are the values of the kinetic energy. The correlation-kinetic energy constitutes a significant fraction of the total energy in the HEC regime as well as for the 2D quantum dot in the LEC regime. Hence, a reduction in dimensionality too leads to a high correlation-kinetic energy. Any low electron density system thus ought to be characterized by both high electron-interaction energy and a high correlation-kinetic energy relative to the total energy.

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