Quantum three-body problem for nuclear physics

Abstract

A brief excursion into the three-body problem in quantum mechanics is presented for graduate students or researchers in nuclear physics. Starting from single-particle coordinates, the three-body Schrödinger equation is systematically transformed into a representation in Jacobi coordinates. Gradient, Laplacian, and kinetic energy operators are explicitly derived using the multivariable chain rule. Faddeev equations are reformulated in hyperspherical coordinates. In all transformations (from single-particle coordinates to Jacobi coordinates, rotation between Jacobi coordinates and from Jacobi coordinates to hyperspherical coordinates) the determinant of the Jacobian matrix is computed to ensure correct transformation of volume elements. The Faddeev equations in hyperspherical coordinates are projected onto a hyperspherical harmonics basis, leading to the coupled hyperradial equations that define the hyperspherical harmonics method.