Spinning Quantum System in an External Magnetic Field

Abstract

In this chapter we will discuss in detail a quantum particle with magnetic moment m = µ _B g J in an external magnetic field B(t) = (math) whose direction (math) is changing periodically. In particular we will consider the case in which the direction of the magnetic field precesses around a fixed axis which we take as the 3-axis (z-axis) of our (laboratory) coordinate frame in space (ℝ³). If the direction rotates slowly (“adiabatically”) this system provides an application of the general ideas developed in Chap. 2. The Schrödinger equation for a magnetic moment in a precessing magnetic field has been solved exactly in [216]. Therefore we need not restrict ourselves to the adiabatic approximation and will obtain the non-adiabatic geometric phase. The latter is also known as the Aharonov-Anandan phase [7, 9] for their pioneering work. With the help of this example we will then introduce in Sect. 3.5 the non-adiabatic geometric phase for a general cyclic evolution.