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Sensitivity analysis for quantum eigenvalues of bound systems
Abstract
The sensitivity of the eigenvalues Ei of a non-integrable Hamiltonian measured by their second differences or second derivatives d2Ei/dλ2 with respect to a coupling parameter λ is studied for the regular and the irregular part of the spectrum as well as for the range of isolated avoided crossings. In the regular part the d2Ei/dλ2 values are expected to be arranged on branches in a plot versus energy: in the avoided-crossing range there should be a drastic increase, and for the irregular part a decrease and a breakdown of the branching is predicted.
This review describes the present state of the quantum chaos problem: The problem of
determining the specific properties of autonomous and nonautonomous quantum systems with
few degrees of freedom whose classical analogs have an unstable (stochastic) motion and also
that of determining the relationships between these properties and the characteristics of the
classical stochastic situation. The criteria for quantum chaos which have been established to date
are examined and compared for autonomous systems. These criteria make use of properties of the
energy spectrum, the wave functions in various representations, the matrices of operators other
than the Hamiltonian, and particular features of the evolution of time-varying states (wave
packets) in such systems. For nonautonomous systems, the conditions for the applicability of
classical dynamics for describing observables and certain specific quantum-mechanical effects
(tunneling through a separatrix and a global quantum resonance) are analyzed.
The nature of the eigenstates of two highly symmetrical two- and three-dimensional quantum oscillators is compared on the basis of their projections on one- and two-dimensional manifolds spanned by separable zero-order states and on the basis of the second derivative E″i of the eigenvalues Ei with respect to a mode coupling parameter λ. Both methods quantitatively lead to the same results in classifying regular states, isolated avoided crossing states and irregular states. The two-dimensional system is the familiar Hénon-Heiles model while in three dimensions an oscillator with tetrahedral symmetry was taken for which the potential energy along the four dissociation channels is identical with the corresponding curves in the Hénon-Heiles model. It is found that there is nearly no irregularity in the spectrum of the 3D model up to dissociation although of the classical trajectories in this energy range propagate chaotically. The 2D system shows a drastic change of the |E″i/Ei| behaviour at the dissociation energy ED indicating a transition from the low-energy regular spectrum to irregularity. In the 3D system no such transition is observed up to energies well above ED. The missing of quantum irregularity when both the dimension and the symmetry of the system are raised, is related to an increase of localization of even the chaotic trajectories found in classical studies. This localization reflects the stabilization of toroids and vague toroids, and possibly the occurrence of low-dimensional classical chaos.
This review describes the present state of the quantum chaos problem: The problem of determining the specific properties of autonomous and nonautonomous quantum systems with few degrees of freedom whose classical analogs have an unstable (stochastic) motion and also that of determining the relationships between these properties and the characteristics of the classical stochastic situation. The criteria for quantum chaos which have been established to date are examined and compared for autonomous systems. These criteria make use of properties of the energy spectrum, the wave functions in various representations, the matrices of operators other than the Hamiltonian, and particular features of the evolution of time-varying states (wave packets) in such systems. For nonautonomous systems, the conditions for the applicability of classical dynamics for describing observables and certain specific quantum-mechanical effects (tunneling through a separatrix and a global quantum resonance) are analyzed.
A subset of the highly excited eigenstates of thiophosgene (SCCl) near the dissociation threshold are analyzed using sensitive measures of quantum ergodicity. We find several localized eigenstates, suggesting that the intramolecular vibrational energy flow dynamics is nonstatistical even at such high levels of excitations. The results are consistent with recent observations of sharp spectral features in the stimulated emission spectra of SCCl
A two dimensional strongly nonharmonic vibrational system with nonlinear intermode coupling is studied both classically and quantum mechanically. The system was chosen such that there is a low lying transition (in energy) from a region where almost all trajectories move regularly to a region where chaotic dynamics strongly dominates. The corresponding quantum system is far away from the semiclassical limit. The eigenfunctions are calculated with high precision according to a linear variational scheme using conveniently chosen basis functions. It is the aim of this paper to check whether the prediction from semiclassical theory, namely that the measure of classically chaotic trajectories in phase space approaches the measure of irregular states in corresponding energy ranges, holds when the system is not close to the classical limit. It is also the aim to identify individual eigenfunctions with respect to regularity and to differentiate between local and normal vibrational states. It is found that there are quantitative and also qualitative differences between the quantum results and the semiclassical predictions. We found a relatively high amount of regularity in the quantum system for energies where almost all classical trajectories move chaotically. This high regularity is manifested via level spacing statistics as well as by a careful study of the nodal behaviour of the eigenfunctions. Another result was not expected from semiclassical predictions: Some of the wave functions are localized and irregular, a behaviour which was not observed in the corresponding classical system.
We review the development of random-matrix theory (RMT) during the last fifteen years. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.
The nearest-neighbor spacing distributions P(S) for the variationally determined eigenvalues of two separable model Hamiltonians (two-dimensional quartic oscillators) are studied as functions of the dimension n of the variational basis. For large basis sizes the spacing histograms for a given energy range can be well fitted by a Poisson distribution, reflecting the regularity of the spectrum of a completely separable Hamiltonian. The decrease of n leads to histograms of qualitatively different spacing (for the same energy range as before), which can be well fitted by Brody distributions Pq(S). The Brody parameter q controls the transition between a Poisson (q=0) and a Wigner distribution (q=1). For large n, q=0 holds. When n falls below a critical value, q=1 results, i.e., Pq(S) turns over into a Wigner distribution. The latter is assumed to characterize a completely irregular spectrum in the classically chaotic energy range of a nonseparable Hamiltonian. The transition between a regular and an irregular spectrum is shown to be induced only by a change of numerical accuracy. We conjecture that similar behavior should also be observed in nonseparable systems.
We study the statistical properties of the distortions of irregular energy spectra when a perturbation parameter is varied; for example, the strength of an external field acting on the bounded quantum system. Three kinds of generalized Calogero-Moser (GCM) classical Hamiltonians are shown to rule the parametric motion of the energy levels in orthogonal, unitary, and symplectic systems. Using these GCM Hamiltonians, we construct a Newtonian theory of ensembles where irregular spectra are correlated with the properties of infinite gases of GCM particles. In this dynamical approach, the results of random matrix theory are recovered. Furthermore, we are able to study parametric properties of irregular spectra such as the level curvature defined by the second derivative of a level energy with respect to the perturbation parameter. We prove that the level curvature density of the orthogonal, unitary, and symplectic systems decreases, respectively, as ||K||-3, ||K||-4, and ||K||-6 for large curvature ||K||. We present numerical results supporting our theoretical analysis and suggesting the universality of the curvature distribution. The relationship of the curvature distribution to the spacing distribution, as well as the possible experimental observation of the curvature distribution, is discussed.
Dynamical behavior of a nonlinearly coupled oscillator system is studied under classical and quantal-classical mixed-mode conditions. Classically, the system displays chaos above an energy threshold. However, upon a partial quantization of the problem within a self-consistent-field formalism, the dynamics becomes highly periodic, pointing out to the smoothing process of the quantum mechanics.
The mixed-mode philosophy of combining classical and quantum degrees of freedom under a single umbrella is employed to study chaotic behavior under quantization. The quantal wave packet is expanded in terms of a set of basis functions. The Jacobi-Hamilton formalism of the time-dependent Schrödinger equation allows the treatment of real and imaginary components of the time-dependent expansion coefficients as coordinates and momenta so that Lyapunov exponents can be calculated. Under the mixed-mode formalism, a two-dimensional nonlinearly coupled oscillator system is partially quantized by letting one of the modes obey classical and the other quantal dynamics. The Lyapunov exponent spectrum of the complete system is obtained and the results are compared with the fully classical ones.
For a model system the distribution of spacings between adjacent levels in specified intervals of the energy spectrum is compared with a semiclassical distribution. This distribution (Berry and Robnik, 1984) depends on the size of the phase space volume filled with irregular trajectories. Good agreement is found between quantal and semiclassical spacing distributions. In particular, the transition from regularity to irregularity observed in the quantum calculation (Haller et al., 1984) is well reproduced by the semiclassical results.
Quantum perturbation theory is used to examine the eigenvalues of a nonseparable Hamiltonian system in the classically regular and irregular regimes. As a function of the perturbation parameter, the eigenvalues obtained by exact (matrix diagonalization) methods undergo an avoided crossing. In the present paper perturbation theory is used as an approximate method to predict the locations of such avoided crossings in energy-parameter space. The sparsity of such avoided crossings in the Hénon–Heiles system is seen to produce regular sequences in the eigenvalues even when the classical motion is predominantly chaotic.
In the regular spectrum of an f-dimensional system each energy level can
be labelled with f quantum numbers originating in f constants of the
classical motion. Levels with very different quantum numbers can have
similar energies. We study the classical limit of the distribution P(S)
of spacings between adjacent levels, using a scaling transformation to
remove the irrelevant effects of the varying local mean level density.
For generic regular systems P(S) = e-S, characteristic of a
Poisson process with levels distributed at random. But for systems of
harmonic oscillators, which possess the non-generic property that the
'energy contours' in action space are flat, P(S) does not exist if the
oscillator frequencies are commensurable, and is peaked about a non-zero
value of S if the frequencies are incommensurable, indicating some
regularity in the level distribution; the precise form of P(S) depends
on the arithmetic nature of the irrational frequency ratios. Numerical
experiments on simple two-dimensional systems support these theoretical
conclusions.
For a model quantum system the distributions of the spacings between
adjacent levels in specified intervals of the energy spectrum are fitted
by the Brody distribution. This analysis displays for the first time the
expected transition from regularity to irregularity, in terms of a
smooth increase of the parameter of the Brody distribution from 0 to 1.
The nature of the eigenstates of two highly symmetrical two- and three-dimensional quantum oscillators is compared on the basis of their projections on one- and two-dimensional manifolds spanned by separable zero-order states and on the basis of the second derivative E″i of the eigenvalues Ei with respect to a mode coupling parameter λ. Both methods quantitatively lead to the same results in classifying regular states, isolated avoided crossing states and irregular states. The two-dimensional system is the familiar Hénon-Heiles model while in three dimensions an oscillator with tetrahedral symmetry was taken for which the potential energy along the four dissociation channels is identical with the corresponding curves in the Hénon-Heiles model. It is found that there is nearly no irregularity in the spectrum of the 3D model up to dissociation although of the classical trajectories in this energy range propagate chaotically. The 2D system shows a drastic change of the |E″i/Ei| behaviour at the dissociation energy ED indicating a transition from the low-energy regular spectrum to irregularity. In the 3D system no such transition is observed up to energies well above ED. The missing of quantum irregularity when both the dimension and the symmetry of the system are raised, is related to an increase of localization of even the chaotic trajectories found in classical studies. This localization reflects the stabilization of toroids and vague toroids, and possibly the occurrence of low-dimensional classical chaos.
The quantal and classical/semiclassical behavior at an isloated avoided crossing are compared. While the quantum mechanical eigenvalue perturbation parameter plots exhibit the avoided crossing, the corresponding primitive semiclassical eigenvalue plots pass through the intersection. Otherwise, the eigenvalues agree well with the quantum mechanical values. The semiclassical splitting at the intersection is calculated from an appropriate Fourier transform. In the quasiperiodic regime, a quantum state near an avoided crossing is seen to exhibit typically more delocalization than the classical state. However, trajectories near the ‘‘separatrix’’ display a quasiperiodic ‘‘transition’’ between two zeroth order classical states.
In the semi-classical limit, the quantal energy spectrum of a dynamical system of N degrees of freedom consists of two parts with strongly contrasting properties: a regular and an irregular part. Properties of vibrational spectra of molecules are predicted.
Statistical analysis of energy level spacings is reported for models of O3, H2O, CO2 and HCN. The levels and their dynamical symmetries are determined by the algebraic approach. Specifically we use Hamiltonians with partial symmetry breaking which therefore have an incomplete set of constants of the motion. Wigner-like nearest-neighbour distributions are found for such non-chaotic systems. In particular, counting of states of the same dynamical symmetry type leads to a Wigner-like distribution. The results suggest that experimental level spacing distributions can serve as a signature for the presence of dynamical (i.e. not necessarily geometrical) symmetries in molecular systems.
A quantum Kolmogorov-Arnol'd-Moser-like theorem is formulated with use of an existence condition of a unique transformation between eigenstates of integrable and nonintegrable Hamiltonians. This condition determines the ability to assign local quantum numbers to eigenstates of nonintegrable Hamiltonians and explains localization phenomena.
Quantum stochasticity (the nature of wave functions and eigenvalues when the short-wave-limit Hamiltonian has stochastic trajectories) is studied for the two-dimensional Helmsholtz equation with "stadium" boundary. The eigenvalue separations have a Wigner distribution (characteristic of a random Hamiltonian), in contrast to the clustering found for a separable equation. The eigenfunctions exhibit a random pattern for the nodal curves, with isotropic distribution of local wave vectors.
At high levels of vibrational excitation it is not realistic to examine the energy levels of molecules state by state. A more coarse grained measure is required. The use of the free energy is illustrated by application to a realistic model Hamiltonian for the stretch vibrations of triatomic molecules, both exact and variational (self-consistent) computations are reported. A transition from a localized to an equipartitioned distribution of energy (as a function of mean excitation energy) is noted and discussed.
The distribution of energy eigenvalues in the irregular spectrum is derived in the semiclassical limit ℏ-->0 by use of a plausible assumption on the spatial distribution of the corresponding eigenfunctions.
Vibrational eigenvalues and eigenfunctions in the strongly coupled regime at intermediate and at high vibrational energy content are characterized in terms of distributions, both of observable and of wave function properties. In this regime, state‐by‐state description in terms of uncoupled modes is both difficult and too detailed; the distributions provide an adequate characterization of the vibrational behavior. We present results from numerical studies of realistic coupled‐vibrational problems; the latter include local‐mode Morse oscillators coupled by the Wilson interaction. The distributions studied are g(ω) (the distribution of nearest‐neighbor energy spacings), G(ω) (the distribution of all energy spacings), π(χ) (the distribution of transition moments), and two different but related wave function expansion coefficient distributions. We find that in the strongly coupled regime, the parameters of the distributions depend only weakly on the Hamiltonian, and, moreover, that simple analytic forms can be found to represent the distributions over a wide range of systems and energies. We conclude that the distributional description of vibrational systems seems potentially very useful and that dynamical elucidation of the distributional parameters seems a desirable aim. Comments are made on the relation to mode mixing, and chaotic‐like behavior.
Semiclassical quantization in the basic, or ‘‘primitive,’’ version often predicts a simpler, more highly degenerate energy level spectrum than is observed. This point is illustrated by remarks on two‐dimensional systems with threefold and higher rotational symmetry and on perturbations of such systems. For example, according to semiclassical quantization essentially all energy levels of a C3 system are at least doubly degenerate. Such ‘‘incorrect’’ degeneracy predictions are to be interpreted as predictions of coalescence of levels in the semiclassical (ℏ→0) limit, coalescence which may manifest itself in real spectra as an excess of near degeneracies. Finally, a consideration on degeneracy rules out one possible solution to the problem of semiclassical quantization of ergodic systems.
The concept of quantum ergodicity is examined empirically through a quantitative study of the eigenfunctions of both the Barbanis and Casati–Ford potentials. By considering the behavior of the nodes of the wave functions and by employing a combination of natural orbital and spectral analyses, many of the higher eigenstates can apparently be classified as ergodic. These particular states seem to correspond quite well not only to classical ergodic trajectories but also to (currently) unquantizable semiclassical wave functions. The relationship between this quantal ergodicity and its classical equivalent is discussed.
In this paper we present the results of a semiclassical investigation and a quantum mechanical study of the bound energy spectrum of the Henon–Heiles Hamiltonian (HHH) and of the Barbanis Hamiltonian (BH). We have derived a simple semiclassical formula for the energy levels E, and for their sensitivity dE/dϵ with respect to the strength ϵ of the nonlinear coupling for the HHH, and established general relations between E and its derivatives dnE/dϵn (n ≥ 1). Numerical quantum mechanical computations of the energy levels were conducted for the HHH and for the BH. The nonlinear coupling constant was adjusted so that for the HHH there will be ∼150 states up to the classical critical energy Ec and ∼300 states up to the dissociation energy ED. The E values were obtained by direct diagonalization using a basis containing 760 states, while the values of dE/dϵ were computed utilizing the Hellmann–Feynman theorem. Good agreement between the semiclassical and the quantum mechanical spectra was observerd well above Ec. These results raise the distict possibility that the semiclassical approxmation for these nonlinear systems does not break down in the vicinity of Ec and that the bound level structure does not provide a manifestation of the classical transition from quasiperiodic to chaotic motion.
Some consequences of classical chaos for semiclassical quantization, the correspondence principle, and molecular energy randomization are outlined. It is suggested that for many of the classically chaotic systems of current interest that the short to intermediate time motion is not actually very much different than for quasiperiodic orbits. This allows introduction of the concept of approximate constants of the motion (or vague tori). The existence of such approximate constants is used to quantize the classical motion, and to set up a framework for definition and calculation of mode-mode energy transfer constants. Preliminary results of calculation of such a rate constant are given. The possible relation of this calculation to quantum dynamics is briefly outlined in the two limits of "large" and "small" values of Planck's constant (an essential consideration for understanding such results).
The classical as well as the quantum dynamics of two-dimensional nonseparable oscillator systems is studied in order to investigate the influence of regular and chaotic motion on the nature of the intermode vibrational energy transfer after local excitation. Representative sets of initial conditions - each element representing a single phase space cell — were chosen. For the classical problem one phase space point per cell was used while for the quantum problem minimum uncertainty wave packets (harmonic oscillator coherent states) with position and momentum expectation values corresponding to the classical values were taken as initial quantum states. For both types of motion information entropies were determined from time averaged probabilities. These entropies are compared with “ergodic limit entropies”. The latter correspond to a statistical preparation of the initial states. Although there is only a weak correspondence between single-trajectory motion and wave packet dynamics, some general predictions about the quantum dynamics can be made from the classical results for a time scale of chemical interest. No sharp transition is found for the wave packet dynamics going from the classically regular to the irregular energy region. However, one can estimate the probability for an arbitrarily prepared local minimum uncertainty wave packet to possess statistical (RRKM) or non statistical (non-RRKM) dynamics from the analysis of a sample of classical trajectories each starting in a phase space cell of the bound area of the Hamiltonian, i.e., the nature of unimolecular reactions can be roughly predicted from the potential energy surface and a sample group of classical trajectories.
A measure of phase space delocalization for the eigenfunctions of regular and irregular eigenstates of a bound hamiltonian is introduced on the basis of a smoothed Wigner distribution function and a phase space graining. This measure, which is an entropy-like quantity, shows a large dispersion as a function of energy above and below the critical energy EC for classically chaotic motion. It is conjectured that many chaotic states also show properties of regular states too.
Anharmonic potentials which are presumed to be classically quasi-ergodic, but which have symmetries leading to degenerate quantum states, fail in the quantum case to transfer energy equivalently among rigorously equivalent phase space locations. This is shown using simple group theory and is illustrated for the case of the Henon-Heiles potential
In plots of eigenvalues of the Schrödinger equation versus a perturbation parameter, many avoided crossings are found in the classically stochastic regime for the system studied. None were observed in the classically quasi-periodic regime. Overlapping avoided crossings are suggested as a mechanism for making the vibrational wavefunction a “statistical” one.
We study particles moving in planar polygonal enclosures with rational angles, and show by several methods that trajectories in the classical phase space explore two-dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply-handled spheres. The quantum mechanics of one such ‘pseudointegrable system’ is studied in detail by computing energy levels using an exact formalism. This system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side L. We find that neighbouring levels avoid degeneracies as L varies, and that the probability distribution for the spacing S of adjacent levels vanishes linearly as S→0 (‘level repulsion’). The Weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density. Oscillatory corrections to the mean level density are given as a sum over closed classical paths; for pseudointegrable systems these closed paths form families covering part of the phase-space invariant surfaces.
In the present paper the classical counterpart of the quantum avoided crossing method for detecting chaos is described using classical (Lie-transform) perturbation theory and a grid of action variables. The results are applied to two systems of coupled oscillators with cubic and quartic nonlinearities. The plots of energy of members of the grid versus the perturbation parameter provide a visual description for predicting the onset of chaos.
Several aspects of the quantal energy spectrum are explored for the Henon–Heiles Hamiltonian system: a striking and initially unexpected continuation of sequences of eigenvalues from the quasiperiodic to the stochastic regime, the origin of large second differences Delta2Ei of eigenvalues arising from variation of a parameter, the comparison of classical and quantal spectra, and a comparison of the "classical" and quantal number of states. In the study of the second differences we find both "crossings" and "avoided crossings" of the eigenvalues. We discuss the importance of overlapping avoided crossings as a basis for a possible theory of "quantum stochasticity".