Perspherical Approaches for Electrons in Atoms and Charged Excitons in Quantum Wells

Instead of following Focks expansion, we solve the Schr?dinger equation for some quantum mechanical many-body systems such as electrons in atoms and charged excitons in quantum wells in a similar way in hyperspherical coordinates by expanding the wave functions into orthonormal complete basis sets of the hyperspherical harmonics (HHs) of hyperangles and generalized Laguerre polynomials (GLPs) of the hyperradius. This leads the equation to a simple recurrence relation of expansion coefficients and solvable. Numerical results obtained explicitly by solving a simple secular equation show good agreement with those obtained through other computationally intensive methods. The eigenenergies and particle correlation in the low-lying states are investigated. The scheme can be improved by considering the factors influencing the rate of convergence in both parts of expansions in wave function with the HHs of hyperangles and the GLPs of hyperradius. By introducing a reselected asymptotic term which includes more structural features, the convergence in the expansion part with the HHs is accelerated, and the use a transformation of the hyperradius can keep the convergence going properly in the expansion part with the GLPs, as demonstrated in calculations for the helium atom in comparison with some other ones. More accurate results were obtained by considering a simple cusp parameter. For charged excitons in two and quasi-two dimensions within a unified framework, numerical results are in good agreement with those obtained through other computationally intensive methods. The mass dependences of eigenenergies and particle correlation in the low-lying states are obtained.