Spin Coherent State And Its Application In The Quantum-classical Correspondence

Spin coherent is a macroscopic quantum state, which has certain quantum-classical correspondence. In present thesis, spin coherent states are introduced in detail and firstly use the spin coherent transformation method to solve geometric phase of the time-dependent evolution system. By spin coherent superposition, we can obtain macroscopic quantum state of corresponding classical orbits. On this base, we discuss quantum-classical correspondence from two aspects:The first one, by using spin coherent state method, we study the correspondence relation between the spatial pattern of probability density of macroscopic quantum wave function and classical periodic orbits. Morever, quantum wave function is expressed as the form of spin coherent states, we obtain that the time-dependent evolution of mechanical quantiy operator expctations on the macroscopic quantum state is consistent with the form of classical dynamic equation, which provides us a frame to study fractional angular momentum. In Chapter 2, we will discuss the probability clouds of the macroscopic wave functions are well localized on classical periodic orbits in two-dimensional anisotropic oscillator with commensurate frequencies. In Chapters 3 and 4, by spin coherent superposition, we construct a macroscopic wave function to study quantum-classical correspondence of the central scalar-potential form A0(r)=-γv/r2μ+2. In Chapter 4, by defining classical spin variables, the classical spin motion equation are obtained. Simultaneously, the quantum-classical correspondence for the spin degree of freedom is also established, which displays exactly the same spin-precession in addition to the orbital motion of central mass. For both closed and open classical orbits, the non-integer angular momentum quantization is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wavefunction. It’s worth mentionging that the SO (Spin-orbital) coupling model in paper exhibits explicitly a pseudo non-Abelian gauge field and the anyon behavior.The other one is the correspondence of geometrical phase in quantum-classical system. In 1984, when Berry studied chaos phenomenon, the Berry phase were accidentally found. In other words, when a quantal system with the multiple parameters is in time-dependent adiabatic evolution, the system will acquire a geometrical phase factor in additional the familiar dynamic phase factor. Even if the system returns to its original state, the additional phase factor is not zero. Shortly after Berry phase discovery, Hannay found analogous with the Berry phase in studying the angle variable of classical integrable systems. An additional angle shift as the system slowly cycles in phase space is acquired, which is called Hannay’s angle. It was later proved by Berry that the geometric phase and Hannay’s angle possess natural relation under semiclassical approximation. The theory of geometrical phase factor is widely applicated in explainning integer quantum Hall effect and abnormal Hall effect et al. In the end, In Chapter 2, in terms of the time-dependent canonical transformation the stationary Lissajous orbits are found along with the Hannay’s angle. By using spin coherent transformation, we discuss that the nonadiabatic Berry phase in the original gauge is found to be (n+1/2) times the nonadiabatic Hannay’s angle.