Excitons Schrodinger equation

Using a particle in a box model with an infinite potential drop at the wall as the boundary condition, and taking into account that the exciton consists of an electron-hole pair, the Schrodinger equation can be solved yielding the energy of the lowest excited state (Brus 1983, 1984 Rossetti et al. 1984), i.e., the lower edge of the conduction band, as... 

In conventional solid-state physics, an exciton is considered as an electron-hole pair separated by a medium with a well-defined dielectric constant, s. The time-independent Schrodinger equation with the inter-carrier Coulomb... 

Presented experimental data reveal that for CdSe/ZnS quantum dots with two ZnS monolayers values follow a monotonous function drastically decaying with the QD core diameter. From the physico-chemical point of view, we conjecture that upon interaction of P with QD surface, the electron wave function may be locally modified (via inductive and/or mesomeric effects [9]) forming a surface local state capable to trap the electron of the photogenerated exciton (Fig. 2A). Thus, we will consider the behaviour of the electron wave function at the interface to the functional pyridyl group of the attached porphyrin. The single-carrier envelope wave functions y/a in a spherical core/shell QD are determined by the Schrodinger equation... 

This equation satisfies the boundary condition of continuity at r= R. The change of energy levels when particles are made small, is schematically shown in Fig. 9.3. A full quantum mechanical treatment of the problem was reported by Brus [13]. Taking into account that the exciton consists of an electron-hole pair, it is advantageous to formulate the Schrodinger equation as follows ... 

There is another reason for degeneracy of excitonic states, being a consequence of the structure of the Schrodinger equation. Indeed, since the Hamiltonian is a self-conjugated operator, wavefunctions l>ko (( = 1,2,..., p), where the star means complex conjugate, as well as wavefunctions kotJ ( = 1,2,..., p),... 

To calculate the effective transition dipole moment we need to know the wave-function of the exciton. In a quantum dot it depends on two interactions (i) the electron and hole confinement potential, which we shall assume to be infinite for r > Ri and zero for r < f i and (ii) the electron and hole Coulomb attraction. For these interactions we have to consider the following characteristic lengths f i - the radius of the quantum dot, and ag - the Bohr radius of an exciton in a macroscopic three-dimensional semiconductor. The problem of solving the two-particle Schrodinger equation for an arbitrary ratio of these lengths is quite difficult but the situation simplifies substantially in two important limiting cases. 

The solution to the spherical Schrodinger equation leads to the energy of the exciton— electron hole pair as ... 

Applying the resolvent operator formalism (25-27), the exciton band levels E) can be obtained as solutions of the Schrodinger equation... 

---