The variational method is used to obtain the ground- and first-excited states for the soft-Coulomb central potential, , characterized by a bias distance d, taken into account as a fixed parameter. Applications are presented for spatially indirect excitons, i.e., photo-generated electron-hole (e-h) bound pairs, where the two charges are kept separated in two different regions of a heterostructure. We consider one- or two-dimensional systems, namely, quantum wires or wells, respectively, and compare the results with numerical calculations obtained by finite-difference diagonalization of the Hamiltonian. An explicit example is given for GaAs-based heterostructures.
References
In these cases, the Schrödinger-like equation determines the envelope function, i.e., the long wavelength part of the true wave function of the crystal. However, often in literature and for our purposes, we do not distinguish between envelope and wave function.
In the 1D case, the ground state has a divergent energy.
Series solutions can still be obtained, e.g., with the Frobenius method; nevertheless, these solutions lack of immediacy, and do not provide a clearer picture of the physical model with respect to fully numerical solutions.
Wolfram Research, Inc., Mathematica 8.0.
It should be noted that the states must be only considered as trial functions, independently of the fact they are solution of some harmonic oscillator (HO) problem: the HO eigenenergies , where , with , have no particular physical meaning for the soft-Coulomb problem here considered. In fact, within the variational method, the energies are defined as expectation values of the Hamiltonian on the states , once the optimization is performed.
For the hole, whose mass tensor is characterized by a strong anisotropy in GaAs, we take the mass along the direction perpendicular to the growth direction of the heterostructure. See Ref. 22 for further details.
