Analytical and numerical solutions for a two-dimensional exciton in momentum space

The effective-mass equation for the quasi-two-dimensional quantum-well exciton takes the form of an integral equation in momentum space. The quadrature method is a good candidate, and has been used to solve the integral equation directly. However, the singular behavior of the continuous-state solutions, and hence the convergence of the quadrature method, has not been examined carefully. In this paper, we first derive the analytical solutions for the pure two-dimensional exciton problem in momentum space and show explicitly that the momentum-space wave functions are the Fourier transforms of the well-known coordinate-space wave functions. Then, we solve the same integral equation numerically by two quadrature methods, one with a constant scaling, and the other with a variable scaling. Numerical results including the energy levels, the oscillator strength of the discrete states, and the enhancement factors of the continuous states are compared with the exact solutions. This comparison provides a general guideline on the accuracy and efficiency of the quadrature method applicable to the case of quasi-two-dimensional excitons when the quantum-size effects are included.