An analytical investigation of a two-electron quantum dot in a quartic anharmonic potential

Abstract

The analytical solutions for two-electron quantum dots (TEQD) are mostly obtained for harmonic confinement. The inclusion of anharmonic terms in the present investigation is certainly a new one for TEQD. The Hamiltonian and hence the Schrodinger equation of TEQD subject to the anharmonic potential is constructed. The Hamiltonian is separable in the centre-of-mass (CM) and relative coordinates. The Schrodinger equations are solved analytically in Cartesian and spherical polar coordinates. The cubic polynomial equation involving the equilibrium position of the CM $R_0$ in spherical polar coordinate is solved by the Cardan method. The TEQD subject to the effective potential in the relative coordinate exhibits a stable minimum. The equilibrium distance between TEQD obeys a polynomial equation of sixth order. The corresponding sixth-order polynomial reduces to two polynomial equations of order three (Cardan equation). Finally, these two Cardan equations are solved analytically to obtain the equilibrium distance between two electrons in TEQD.