Abstract With this paper Grimm and Storer introduced importance sampling to the field of quantum Monte Carlo. They had earlier used numerical quadrature for estimating the integrals required in evaluating energies in a diffusion-like path integral iteration scheme for solving the Schrödinger equation.0 This was extended to larger numbers of particles and higher dimensions with use of Monte Carlo methods. In conjunction with this they recognized they could take advantage of prior knowledge of the general form of a wavefunction to obtain more accurate estimates of the eigenvalue Eo .
A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials.
A different integral form of the Schr\"odinger equation and some boundedness conditions of the wave functions are proved, which exhibit the origin of some divergences in quantum mechanics. The iterations from the integral equation and the boundedness conditions lead to a method for avoiding the divergence difficulties. The result is extended to the time-dependent and the multidimensional cases. Examples of an electron in the Wigner spherical ``box'' with a
1/r perturbed potential, some heliumlike ions in the ground state, and the Stark effect of the hydrogen atom show that this different perturbation method can give rational multiorder corrections of the wave function and energy.