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Acta Physica Sinica

ALGEBRAIC DYNAMICS AND LINEAR NONAUTONOMOUS QUANTUM SYSTEMS WITH SU(2) DYNAMICAL GROUP

Zuo Wei, Shun-Jin Wang

Published Jan 1, 1995• 0 citations
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Abstract

The exact solution and the invariant Cartan operator of the linear nonautono-mous system with SU(2) dynamical group are obtained with the method of algebraic dynamics. The profound quantum-classical correspondence of the solutions is exhibited clearly: the quantum solutions of the system are determined by that of the corresponding classical equations. The exact solutions are employed to calculate the inversion probability of a neutrino in a time-dependent magnetic field and the-particle spin polarization in beam dynamics. Nonadiabatic Berry's phase for a periodic system is also calculated.

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The exact solution and the invariant Cartan operator of the linear nonautono-mous system with SU(2) dynamical group are obtained with the method of algebraic dynamics. The profound quantum-classical correspondence of the solutions is exhibited clearly: the quantum solutions of the system are determined by that of the corresponding classical equations. The exact solutions are employed to calculate the inversion probability of a neutrino in a time-dependent magnetic field and the-particle spin polarization in beam dynamics. Nonadiabatic Berry's phase for a periodic system is also calculated.
We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.
The nonadiabatic Berry's phase is investigated for a quantum system with a dynamical semisimple Lie group within the framework of the generalized cranking approach. An expression for nonadiabatic Berry's phase is given, which indicates that the nonadiabatic Berry's phase is related to the expectation value of Cartan operators along the cranking direction in group space, and that it depends on (i) the geometry of the group space, (ii) the time evolution ray generated by the Hamiltonian (i.e., by the dynamics) in some irreducible representation Hilbert space, and (iii) the cranking rate. The expression also provides a simple algorithm for calculating the nonadiabatic Berry's phase. The general formalism is illustrated by examples of SU(2) dynamic group.
Based on the homotrinuclear spin cluster having SU(2)⊗SU(2) symmetry with twofold orbital degeneracy τ = 1/2) and the SU(2) algebraic structures of both ŝ and τ subspaces in the external magnetic field, we calculate exactly the non-adiabatic energy levels and the cyclic and non-cyclic non-adiabatic geometric phase of the homotrinuclear spin cluster by making use of the method of algebraic dynamics. The solution will show that the Berry phase is much influenced by the parameters N = γs/γτ (γs and γτ are the magnetic momentums of ŝ and τ subspaces, respectively) in addition to ω/Ω in a rotating magnetic field. The change of the Berry phase in the basis state of the system is demonstrated from the changing diagram.
We find a relationship between unitary transformations of the dynamics of quantum systems with time-dependent Hamiltonians and gauge theories. In particular, we show that the nonrelativistic dynamics of spin-12\frac{1}{2}21​ particles in a magnetic field Bi(t){B}^{i}(t)Bi(t) can be formulated in a natural way as an SU(2) gauge theory, with the magnetic field Bi(t){B}^{i}(t)Bi(t) playing the role of the gauge potential Ai{A}^{i}Ai. The present approach can also be applied to systems of n levels with time-dependent potentials, U(n)U(n)U(n) being the gauge group. This geometric interpretation provides a powerful method to find exact solutions of the Schr\"odinger equation. The root of the present approach rests in the Hermiticity property of the Hamiltonian operators involved. In addition, the relationship with true gauge symmetries of n-level quantum systems is discussed.
The time-dependent Schr\"odinger equation for a spin particle in a rotating magnetic field is solved analytically by the cranking method, and the exact solutions are employed to study the nonadiabatic Berry's phase. An alternative expression for Berry's phase is given, which shows that Berry's phase is related to the expectation value of spin along the rotating axis and gives Berry's phase a physical explanation besides its gauge geometric interpretation. This expression also presents a simple algorithm for calculating the nonadiabatic Berry's phase for Hamiltonians that are nonlinear functions of the SU(2) generators. It is shown that nonadiabaticity alters the time evolution ray and in turn changes its Berry's phase. For the SU(2) dynamical group, the nonadiabatic effect on Berry's phase manifests itself as spin alignment (a phenomenon in nuclear physics), and spin-alignment quantization (observed recently in high-spin nuclear physics) is related to Berry's-phase quantization.