Hyperspherical method

A method for describing few-particle systems with coordinates that separate overall size from shape. It converts their motion into coupled equations in a hyperradius and angular variables.

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Hyperradius: A radial coordinate measuring the overall size of a few-particle configuration relative to its center of mass. It isolates collective size from the angular variables describing shape.
Configuration space: The space of all possible positions or configurations of a physical system. Few-particle coordinates form a multidimensional space whose geometry motivates hyperspherical coordinates.
Three-body problem: The problem of predicting the motion or quantum states of three interacting bodies. Three particles are the standard setting where the method exposes coupled shape and size dynamics.
Hyperspherical coordinates: A higher-dimensional coordinate system consisting of a radial coordinate and angles on a hypersphere. These are the variables at the mathematical core of the method.
Hyperspherical harmonics: Angular basis functions on a sphere in a multidimensional configuration space. They expand the shape dependence of wavefunctions in the method.
Center-of-mass frame: A reference frame in which the system’s center of mass is at rest. Removing center-of-mass motion leaves the relative configuration described by the method.
Efimov effect: A universal phenomenon in which three particles can form an infinite sequence of bound states near resonant interactions. Its long-range three-body attraction appears transparently in hyperspherical channels.
Cartesian coordinates: A coordinate system that specifies positions through perpendicular component axes. They keep particle positions direct but do not separate collective size from shape.
Adiabatic hyperspherical representation: A formulation that solves angular motion at fixed hyperradius and couples the resulting channels through radial motion. It turns the method’s coupled problem into hyperradial channel equations.
Spherical coordinates: A coordinate system that represents position using a radius and angular variables. The hyperspherical construction generalizes this radial-and-angular split to higher dimensions.