Abstract. We present a review of the quantum three-body problem, with emphasis on the different methodologies, different three-body atomic systems and their historical interest. With the review as the background, a more recently proposed non-variational, kinetic energy operator approach to the solution of quantum three-body problem is presented, based on the utilization of symmetries intrinsic to the kinetic energy operator, i.e., the three-body Laplacian operator with the respective masses. Through a four-step reduction process, the nine dimensional problem is reduced to a one dimensional coupled system of ordinary differential equations, amenable to accurate numerical solution as an infinite-dimensional algebraic eigenvalue problem. A key observation in this reduction process is that in the functional subspace of the kinetic energy operator where all the rotational degrees of freedom have been projected out, there is an intrinsic symmetry which can be made explicit through the introduction of Jacobi-spherical coordinates. A numerical scheme is presented whereby the Coulomb matrix elements are calculated to a high degree of accuracy with minimal effort, and the truncation of the linear equations is carried out through a systematic procedure. The resulting matrix equations are solved through an iteration process. Numerical results are presented for (1) the negative hydrogen ion H − , (2) the helium and heliumlike ions (Z = 3 ∼ 6), (3) the hydrogen molecular ion H + 2 , and (4) the positronium negative ion Ps − . Up to thirteen-significant-figure accuracy is achieved for the ground state eigenvalues when double precision programming is used. Comparison with the variational and other approaches shows our ground state eigenvalues to be comparable, generally with less decimal digits than the variational results, but can yield highly accurate wavefunctions as by-products. Results on low-lying excited states and their wavefunctions are obtained simultaneously with the ground state properties, some at accuracies not achieved by other methods. In particular, for the doubly excited state 3 P e of H − and the 1,3 P e states of helium, some results are obtained for the first time. Also, we have calculated fourteen H + 2 excited states, up to its dissociation level. Analysis of the wavefunction characteristics, especially in relation to the electronelectron correlation effects, are presented. A significant advantage of the kinetic energy operator approach is its general applicability to different three-body systems, with only the charges, masses, and the symmetry of the desired state as the required inputs. Potential applications of the present approach to scattering and other problems are noted.
