The concepts of integrability, non-integrability and chaos in quantum mechanics are examined, and it is indicated that they all are sensibly definable only in connection with the corresponding properties of their classical analogues. The concrete examples concern the quantum and classical properties of dynamic systems on SU3 algebra and classically integrable but quantum nonintegrable system with two degrees of freedom.
The concepts of integrability, non-integrability and chaos in quantum mechanics are examined, and it is indicated that they all are sensibly definable only in connection with the corresponding properties of their classical analogues. The concrete examples concern the quantum and classical properties of dynamic systems on SU3 algebra and classically integrable but quantum nonintegrable system with two degrees of freedom.
In this paper we discuss the concepts of quantum integrability and nonintegrability. Based on the concept of a complete set of commuting observables and the Hilbert-space structure of a quantum system, the definitions are given for the quantum-dynamical degrees of freedom and quantum phase space from which the quantum integrability is defined. A criterion for quantum integrability then emerges; the system is integrable if it possesses dynamical symmetry. Breaking of dynamical symmetry is connected with the nonintegrability of systems and thus is the inherent mechanism of chaotic motion. A number of examples are discussed.
Based on the concepts of integrability and nonintegrability of a quantum system presented in a previous paper [Zhang, Feng, Yuan, and Wang, Phys. Rev. A 40, 438 (1989)], a realization of the dynamics in the quantum phase space is now presented. For a quantum system with dynamical group scrG and in one of its unitary irreducible-representation carrier spaces ${\mathit{gerh}}_{\mathrm{\ensuremath{\Lambda}}}$, the quantum phase space is a 2${\mathit{M}}_{\mathrm{\ensuremath{\Lambda}}}$-dimensional topological space, where ${\mathit{M}}_{\mathrm{\ensuremath{\Lambda}}}$ is the quantum-dynamical degrees of freedom. This quantum phase space is isomorphic to a coset space scrG/scrH via the unitary exponential mapping of the elementary excitation operator subspace of scrg (algebra of scrG), where scrH (\ensuremath{\subset}scrG) is the maximal stability subgroup of a fixed state in ${\mathit{gerh}}_{\mathrm{\ensuremath{\Lambda}}}$. The phase-space representation of the system is realized on scrG/scrH, and its classical analogy can be obtained naturally. It is also shown that there is consistency between quantum and classical integrability. Finally, a general algorithm for seeking the manifestation of ``quantum chaos'' via the classical analogy is provided. Illustrations of this formulation in several important quantum systems are presented.
In this Letter, we describe the connection between the classical concepts of nonintegrability and chaos and the quantum concept of dynamical symmetry breaking. The existence of unbroken dynamical symmetry implies integrability of the mean-field motion in a quantum phase space, defined as a symplectic coherent-state parametrization of the coset space of the overall dynamical group. Broken dynamical symmetry leads to nonintegrability, and thus chaotic solutions to Hamilton's equations in the quantum phase space. We illustrate the general ideas with results obtained for a model of two coupled spins.
It is known that the development and deep understanding of modern interaction theory and classical mechanics are made through geometry and symmetry. Yet, quantum mechanics which was regarded to be the microscopic theory of classical mechanics and achieved the crowning success in interpreting the entire microscopic world was developed purely from algebraic methods. In this thesis, we will study the geometry and dynamical symmetry in quantum systems, from which the question of integrability and chaos are explicitly addressed. First of all, the quantum dynamical degrees of freedom and quantum integrability are precisely defined and the inherent geometrical structure of quantum systems is explored from the fundamental structure of quantum theory. Such a geometrical structure can provide a framework to simultaneously build quantum and classical mechanics. The quantum-classical correspondence is then explicitly deduced. The dynamics of quantum system before it reaches the classical limit is formulated. Thus, the classical chaos is proven to be a special limiting phenomena of quantum systems and the dynamics before the system reaches its classical chaos is explored. The latter is the first step to seek the quantum manifestation of chaos. The relationship between integrability and dynamical symmetry are studied and some universal properties are discovered: a dynamical system (both quantum and classical) is integrable if it possesses a dynamical symmetry. Chaos will occur if the system undergoes a dynamical symmetry breaking and is accompanied by a structural phase transition. Thus, the concept of dynamical symmetry can be used to predict the general behaviors of a system. The theoretical underpinnings developed in this thesis are verified by many basic quantum mechanical examples. In the second part, we have applied the theory to construct the geometrical description for fermionic systems and then use this general formulation to study the geometrical properties, integrability and chaos in nuclear collective motions. The study of geometry for the fermion systems clearly show how the pure quantum properties (e.g. Pauli principle) are exhibited in geometry and how these geometrical manifestations can interpret experimental facts in nuclear collective motion (e.g. the satiation of nuclear deformation and narrow oblate window). The dynamical symmetry and dynamical symmetry breaking in nuclear collective motion are extensively studied.
The subject of the four manuscripts which make up this dissertation is the concept of integrability in quantum mechanical systems with two degrees of freedom, including the relation between quantum integrability and classical integrability in such systems. In the first manuscript, I consider a two-spin model both classically and quantum mechanically. In the six-dimensional parameter space of this model, classical integrability is satisfied on a five-dimensional hypersurface, and level crossings occur on four-dimensional manifolds that are completely embedded in the integrability hypersurface except for some lower-dimensional sub-manifolds. Under mild assumptions, the classical integrability condition can be reconstructed from a purely quantum mechanical study of level degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix. In the second manuscript, I analyze the functional dependence [special characters omitted] of the Hamiltonian operator on the action operators for an integrable quantum system, and I compare it with the corresponding functional relationship H (p1, q1; p2, q 2) = HC(J1, J2) in the classical limit of that system. It is demonstrated that the functional dependence is not the same quantum mechanically and (semi-)classically. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence may be non-uniform. This analysis is carried out for an integrable one- parameter two-spin model. Additional results presented for the (integrable) circular billiard model exhibit similar quantum corrections to the predictions of semiclassical quantization. In the third and the fourth manuscripts, I investigate the relations between symmetry, integrability, and the assignment of quantum numbers to eigenstates for two-spin and spin-boson models. I calculate quantum invariants in the form of expectation values for selected operators and monitor their dependence on the Hamiltonian parameters along loops within, without, and across the integrability region in parameter space. I find clear-cut signatures of integrability and nonintegrability in the observed traces of quantum invariants evaluated in invariant Hilbert subspaces. The results support the notion that quantum integrability depends on the existence of action operators as constituent elements of the Hamiltonian.
Does quantum chaos exist?If it exists, what is it?If it does not, what was it supposed to have been?Why should we associate chaos with a "third revolution in physics" if it fails to be expressible in terms of quantum mechanics, our most fundamental theory of physical reality?Is there something wrong with quantum mechanics?Or is chaos merely a mathematical construct relevant only for models in classical mechanics?Is the human mind doomed to interpret and understand quantum mechanics in classical terms?The list of unanswered questions in quantum chaos research itself shows symptoms of the phenomenon it attempts to grasp. 1 How do we recognize classical Hamiltonian chaos?We have dealt with this question in our two previous columns.2,3 In Part I we discussed the implications of integrability and non-integrability for the phase-space trajectories of classical Hamiltonian systems with two degrees of freedom.We described the method of Poincaré surfaces of section as a convenient and striking discriminant between the two possibilities.In Part II we searched for an alternate mode of representation for classical Hamiltonian systems, a mode that is an equally powerful indicator of chaos, but one that can be more directly translated into quantum mechanics than the Poincaré map.We proposed a representation based on the construction of classical invariants via time averages of dynamical variables, a representation that can be employed under very general circumstances including integrable and nonintegrable models.In this column we shall construct quantum invariants, demonstrate the impact of non-integrability on these quantities, and discuss their properties in relation to their classical counterparts.In a future column we intend to introduce quantum Poincaré surfaces of section for stationary states and bring full circle our survey of quantum and classical Hamiltonian chaos.Consider a quantized integrable model.Its eigenstates are labeled by quantum numbers.There are as many quantum numbers as there are degrees of freedom.Each quantum number represents a quantized action.In the still evolving language of quantum chaos, quantum numbers are said to have been "lost" in the transition from a quantum system whose classical counterpart is integrable to one whose counterpart is chaotic.The broken, or lost, tori of classical chaos become lost quantum numbers of quantum chaos.As we shall see, the loss of quantum numbers is vividly mirrored in the behavior of the quantum invariants.We begin by describing the construction of the quantum invariants for a