The dynamical variational approach based on geometrical quantization is demonstrated to be capable in describing the most important quantum mechanical quantities. In particular, the consistent prescription for calculating the transition probabilities is presented. For several reasons, the method is expected to be better under control than the semiclassical methods in treating the systems whose classical counterparts are chaotic. The formal considerations are illustrated using an exactly solvable SU(3)-spin system
The dynamical variational approach based on geometrical quantization is demonstrated to be capable in describing the most important quantum mechanical quantities. In particular, the consistent prescription for calculating the transition probabilities is presented. For several reasons, the method is expected to be better under control than the semiclassical methods in treating the systems whose classical counterparts are chaotic. The formal considerations are illustrated using an exactly solvable SU(3)-spin system
Einstein (1917) appreciated early on that within the “old” pre-1925/26 quantum mechanics absence of integrability is a serious obstacle for the quantization of classical systems. Therefore, in retrospect not surprisingly, the quantization problem was not adequately solved until the advent of the “new” quantum mechanics by Heisenberg, Born, Jordan and Schrödinger. The new quantum mechanics did not rely at all on the notion of classical paths, and this way, unwittingly, sidestepped the chaos problem. Within the framework of the new theory, any classical system can be quantized, including classically chaotic systems. But while the quantization of integrable systems is straightforward, the quantization of classically chaotic systems, even today, presents a formidable technical challenge. This is especially true for quantization in the semiclassical regime, where the quantum numbers involved are large. In fact, efficient semiclassical quantization rules for chaotic systems were not known until Gutzwiller (1971, 1990) intoduced periodic orbit expansions. Gutzwiller's method is discussed in Section 4.1.3 below. It is important to emphasize here that the existence of chaos in certain classical systems in no way introduces conceptual problems into the framework of modern quantum theory, although, let it be emphasized again, chaos came back with a vengeance from the “old” days of quantum mechanics. Even given all the modern day computer power accessible to the “practitioner” of quantum mechanics, chaos is the ultimate reason for the slow progress in the numerical computation of even moderately excited states in such important, but chaotic problems as the helium atom.
It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.
A new method for the quantization of classical Hamiltonian systems is presented. This method is based upon the correspondence between the Liouville formulation of classical mechanics and the Liouville–von Neumann formulation of quantum mechanics. It does not distinguish between integrable and nonintegrable systems, and consequently, is equally applicable to both types of systems. Further, it treats the indistinguishability of identical particles correctly, and thus, the semiclassical eigenstates have the correct symmetry properties. Application of the method is illustrated by a series of examples. The results are in excellent agreement with quantum mechanics and represent an improvement over results obtained using the uniform semiclassical approximation.
Abstract The conventional variational method is reformulated within a geometrical approach to quantum mechanics, both for finite and infinite dimensional Hilbert spaces. This geometrical setting allows us to deal not only with approximated eigenvalues and eigenstates, but also with approximated Schrödinger equations by means of Hamiltonian evolution equations associated with expectation value functions. We shall consider some specific examples where we extend our approach also to density matrices.
We present a formalism for simulating quantum dynamics of lattice spin-1 systems by first introducing local hidden variables and then doing semiclassical (truncated Wigner) approximation in the extended phase space. In this way, we exactly take into account the local on-site Hamiltonian and approximately treat spin-spin interactions. In particular, we represent each spin with eight classical SU(3) variables. Three of them represent the usual spin components and five others are hidden variables representing local spin-spin correlations. We compare our formalism with exact quantum dynamics of fully connected spin systems and find very good agreement. As an application, we discuss quench dynamics of a Bose-Hubbard model near the superfluid-insulator transition for a 3D lattice system consisting of 1000 sites.
A method is proposed to obtain the dynamics of a system which only makes use of the group law. It incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics: The classical or quantum character of the theory is related to the choice of the group, avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems, i.e., the classical limit. The method is applied to the free-particle dynamics and the harmonic oscillator.