Eigenvalue problem of the Schrödinger equation via the finite-difference time-domain method

We present a very efficient scheme to calculate the eigenvalue problem of the time-independent Schrödinger equation. The eigenvalue problem can be solved via an initial-value procedure of the time-dependent Schrödinger equation. First, the time evolution of the wave function is calculated by the finite-difference time-domain method. Then the eigenenergies of the electron system can be obtained through a fast Fourier transformation along the time axis of the wave function after some point. The computing effort for this scheme is roughly proportional to the total grid points involved in the structure and it is suitable for large scale quantum systems. We have applied this approach to the three-dimensional GaN quantum dot system involving one million grid points. It takes only 7 h to calculate the confined energies and the wave functions on a standard 2-GHz Pentium 4 computer. The proposed approach can be implemented in a parallel computer system to study more complex systems.