Analytical solutions for two electrons in an oscillator potential and a magnetic field

Two electrons subject to the Coulomb interaction and confined by an anisotropic harmonic potential can be viewed in two ways. One is the “Hooke’s atom,” much used in density functional theory (DFT), but with the addition of an external magnetic field. The other is as a heliumlike three-dimensional quantum dot. Though some results on the systems are known in both the DFT and quantum dot literature, exact analytical solutions have been lacking for nonzero anisotropy (nonzero $B$). For certain specific confinement strengths, we develop such solutions in closed form, hence include electron exchange and correlation exactly. As with the zero-field Hooke’s atom, those solutions can be the ground or excited states.

Figure 1

Confinement strengths amenable to analytical solution of Eq. (2). Panels (a)–(c) are for $m=0$, 1, 2, respectively. The 10 symbols (hexagon, square, up triangle, diamond, down triangle, circle, left triangle, plus sign, right triangle, and $x$ mark) correspond to integers 1, 2, …, 10 that are the highest order of the polynomial factor of the relative motion wave functions in $z$ for 1D, in $\rho$ for 2D, in $r$ for the 3D HA $(B=0)$, and in $t$ for the QD. For the 3D HA in $B\ne 0$, those symbols stand for the values of $(N_z+{\pi }_z)$ in, for example, Eq. (12). For the spherical HA, only ${\pi }_z=0$ is included; notice that its odd parity $({\pi }_z=1,m)$ and even parity $({\pi }_z=0,m+1)$ states are degenerate. For the 2D case, ${\omega }_z=\infty$ has been shifted to ${\omega }_z=1$. For the 1D case, ${\omega }_{\rho }=\infty$ has been shifted to ${\omega }_{\rho }=1$.