Geometry of adiabatic changes. General analysis

The concept of adiabatic change and related notions are discussed. As a representative example, the evolution of a neutron spin in a precessing magnetic field is described briefly. The general setting of the equations for Dirac's evolution coefficients is discussed from the geometric phase point of view. An exactly solvable model of nutation is exhibited and the properties of the solutions are analyzed to reveal the holonomic structure of the problem. The corresponding expression for the geometric phase differs nontrivially from the corresponding expression in the well-known case of the precessing field. In addition, this geometric phase has an imaginary part which completes the picture of spin evolution in a nutation mode. The approach proposed for nutation is used to reexamine the twisted Landau-Zener problem. … Thusγ _n(C) is given by a circuit integral in parametric space and is independent of how the circuit is traversed (provided of course that this is slow enough for the adiabatic approximation to hold).M. V. Berry To the memory of my father.