Progress in analytical solution of Schrödinger equation for helium atom

In my talk I review the progress in the process of analytical solution of Schrödinger equation for the helium atom. The problems of two correlated electrons in a central field of nucleus, the simplest unsolved problem of... [ view full abstract ]

In my talk I review the progress in the process of analytical solution of Schrödinger equation for the helium atom. The problems of two correlated electrons in a central field of nucleus, the simplest unsolved problem of quantum mechanics, attracted a lot of attention over the last 80 years. Most of these results are almost completely unknown in the quantum community. Implicit analytical form of the ground state wave function for helium was proposed by Fock in 1954 [1]. Explicit determination of the angular Fock coefficients turned out to be a complex task; only few of them are found up to date [1–3]. The Fock series can be considered as an extended Taylor series of the exact wave function around the triple coalescence point. It is important to stress that the Fock series represents all the L=0 solutions to the Schrödinger equation, including the divergent ones. The identification of the physically meaningful solutions proceeds via studying the behavior of the Fock series in the limit of large electron-nucleus separations. The current talk will show how the Fock series naturally emerges from a homogeneity expansion proposed by Hylleraas and further developed by our group. In particular, it will show how the logarithmic terms appear in low-order Fock coefficients in order to guarantee physical behavior (continuity and finiteness) of the resulting wave function. We will address also the question of Fock expansion for helium in states with non-zero angular momentum.