Excited states of the Gaussian two-electron quantum dot

Abstract

We consider the 1s2s 1,3 S states of the two-electron three-dimensional quantum dot with a Gaussian one-body potential, −V0 exp(−λr 2 ). For a single electron, a simple scaling relation allows the reduction into a one-parameter problem in terms of V 0 λ . However, for the two-electron system, the interelectronic repulsion term, 1 r 12 , frustrates this simple scaling transformation, so we face a genuine two-parameter system. We pay particular attention to the location and nature of the critical well-depths, at which the binding energy of the second electron vanishes. Several observations are noteworthy: For all λ, the triplet critical well-depth is lower than that in the singly excited singlet state. Hence, there exists a finite range of well-depths for which the triplet is bound and the singlet is not, a feature that can possibly be applied in some device. Above its critical well-depth, the triplet state energy is always lower than that of the singly excited singlet. Both well-depths are considerably higher than the critical well-depth in the ground state. The expectation value of the interelectronic repulsion is always lower in the triplet, like the harmonic quantum dot but unlike He-like atoms, the two-particle Debye (Yukawa) atom, or the confined He atom. In the infinite well-depth (V0) limit, keeping the well-width 1 λ constant, the energies and other expectation values of the bound states of the two-electron Gaussian quantum dot approach those of a non-interacting harmonic two-electron system.

Dispersion and entropy-like measures of multidimensional harmonic systems. Application to Rydberg states and high-dimensional oscillators

“…which does not dependence on the oscillator strength ω as expected because of the homogenous property of the oscillator potential [70]. Moreover, for k = 2 it gives the familiar relation…”

The weakly bound states in Gaussian wells: From the binding energy of deuteron to the electronic structure of quantum dots

“…This can be achieved by selecting a reference frame such that $$$ {\theta}_2=\pi /2 $$$ as shown in Figure 2, reducing the wave function effective dependence to three coordinates: $$$ {r}_1 $$$, $$$ {r}_2 $$$, and $$$ {\theta}_1 $$$, being the later the angle between the two vectors. The Hamiltonian () does not admit an analytical solution, and similar to the one body case, it possesses a finite number of bound states which have been explored extensively in [50] within the LMM approach. Under the rescaling $$$ {r}_i\to {\lambda}^{1/2}{r}_i $$$, the Hamiltonian () transforms as $$$$ \frac{1}{\lambda}\hat{H}=-\frac{\hslash }{2m}\left({\nabla}_1^2+{\nabla}_2^2\right)-\frac{V_0}{\lambda}\left({e}^{-{r}_1^2}+{e}^{-{r}_2^2}\right)+\frac{e^2}{\sqrt{\lambda }{r}_{12}}.$$…”

“…This can be achieved by selecting reference frame such that θ 2 ¼ π=2 as shown in Figure 2, reducing the wave function effective dependence to three coordinates: r 1 , r 2 , and θ 1 , being the later the angle between the two vectors. The Hamiltonian (21) does not admit an analytical solution, and similar to the one body case, it possesses a finite number of bound states which have been explored extensively in [50] within the LMM approach. Under the rescaling r i !…”

Two trapped particles interacting by a finite-ranged two-body potential in two spatial dimensions

“…The study of Exact solution of harmonically trapped two ultra cold spin-0 bosons in 2D interacting via finite range two body potential modelled by a Gaussian potential, has been shown to put vast interest in the last few decades in the field of optical lattices, quantum information and entanglement has been discuss for the two identical particles [4][5][6]. Manybody physics is rather complex when atom-atom interaction is considered.…”