Variational Methods in the Quantum Mechanical Three-Body Problem with a Coulomb Interaction

The variational Rayleigh–Ritz method for bound states in nonrelativistic quantum mechanics is formulated and the mathematical foundations of the method are discussed. A review of the most frequently used methods for constructing the Ritz variational basis is given on the example of the helium atom. Numerous applications of variational methods to solving the quantum mechanical three-body bound state problem are considered, and a comparison of the most accurate calculations for various physical systems is given. As a generalization of the variational Rayleigh–Ritz method to “quasi-bound” states (resonances), the complex coordinate rotation (CCR) method and its application to calculating various characteristics of resonances are considered.