Numerical Calculation of the Wave Functions and Energies of the1S1and2S3States of Helium

An exact iteration method for obtaining solutions to the eigenvalue problems of quantum mechanics is used as the basis for developing a numerical iteration scheme for the approximate solution of such problems. The connection between an approximate analytic iteration method and the standard variational method is made and the former method is applied to the $1{}_{}{}^1{}_{}{}^{}S$ state of He. The wave functions so determined are linear combinations of products of hydrogen-like wave functions. The best value of the energy obtained with twenty parameters is $E\left(1\mathrm{}{}_{}{}^1{}_{}{}^{}S\right)=-2.900938\mathrm{}$ au. By using the theory of Gaussian quadrature and least-squares approximation, a systematic transition from an exact iteration method to the numerical iteration method can be made. The resulting numerical scheme is applied to the $1{}_{}{}^1{}_{}{}^{}S$ and $2{}_{}{}^3{}_{}{}^{}S$ states of He. The energies obtained are $E\left(1\mathrm{}{}_{}{}^1{}_{}{}^{}S\right)=-2.903443\mathrm{}$ au. and $E\left(2\mathrm{}{}_{}{}^3{}_{}{}^{}S\right)=-2.174823\mathrm{}$ au. The $2{}_{}{}^3{}_{}{}^{}S$ wave function yields a ${\mathrm{He}}^3$ hyperfine splitting $\nu =6664\mathrm{}$ Mc/sec, which is lower than the experimental value by about 1%. The wave functions obtained are expressible in both the coordinate and momentum representations.