Variational method using basis functions

The Gaussian, exponential and Laguerre basis functions are examined in a variational calculation of energies and wavefunctions. The Laguerre basis set is already orthonormal and complete, but the Gaussian and exponential basis sets are not orthonormal. We used the linear and Coulomb potentials to test these basis functions. Calculations are performed in both position and momentum space. We also present the results with relativistic kinematics in the momentum space calculation. The Gram-Schmidt procedure is used to orthonormalize the Gaussian and exponential basis sets before using them in the calculations. We show that in the case of a pure linear potential, the orthonormal basis constructed from the Gaussian functions performs much better than the exponential and Laguerre basis for the same number of orthogonal functions. For Coulomb-like potentials, the exponential basis performs better than the other two for the same number of basis functions. The advantage of using these simple basis functions is that for the potentials that we examined, one can approach the lower bound of the low-lying states with very few basis functions.