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Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?
Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.
Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, meaning there is sensitive dependence on initial conditions. A metaphor for this behavior is that a butterfly flapping its wings in China can cause a hurricane in Texas.
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations.
In seeking to address the basic question of quantum chaos, several approaches have been employed:
During the first half of the twentieth century, chaotic behavior in mechanics was recognized (as in the three-body problem in celestial mechanics), but not well understood. The foundations of modern quantum mechanics were laid in that period, essentially leaving aside the issue of the quantum-classical correspondence in systems whose classical limit exhibit chaos.
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists, as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics to astronomical objects, such as stars and planets, to produce ephemeris data.
Questions related to the correspondence principle arise in many different branches of physics, ranging from nuclear to atomic, molecular and solid-state physics, and even to acoustics, microwaves and optics. Important observations often associated with classically chaotic quantum systems are spectral level repulsion, dynamical localization in time evolution (e.g. ionization rates of atoms), and enhanced stationary wave intensities in regions of space where classical dynamics exhibits only unstable trajectories (as in scattering).
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which studies the atom as a whole, including its electrons.
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and the processes by which these arrangements change. This comprises ions, neutral atoms and, unless otherwise stated, it can be assumed that the term atom includes ions.
Molecular physics is the study of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics. Its most important experimental techniques are the various types of spectroscopy; scattering is also used. The field is closely related to atomic physics and overlaps greatly with theoretical chemistry, physical chemistry and chemical physics.
In the semiclassical approach of quantum chaos, phenomena are identified in spectroscopy by analyzing the statistical distribution of spectral lines and by connecting spectral periodicities with classical orbits. Other phenomena show up in the time evolution of a quantum system, or in its response to various types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular amplitude distributions.
Spectroscopy is the study of the interaction between matter and electromagnetic radiation. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, by a prism. Later the concept was expanded greatly to include any interaction with radiative energy as a function of its wavelength or frequency, predominantly in the electromagnetic spectrum, though matter waves and acoustic waves can also be considered forms of radiative energy; recently, with tremendous difficulty, even gravitational waves have been associated with a spectral signature in the context of the Laser Interferometer Gravitational-Wave Observatory (LIGO) and laser interferometry. Spectroscopic data are often represented by an emission spectrum, a plot of the response of interest, as a function of wavelength or frequency.
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state. In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies is governed by the principles of classical mechanics. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by Newton's laws of motion. These principles can also be equivalently expressed more abstractly by Hamiltonian mechanics or Lagrangian mechanics.
The amplitude of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the variable's extreme values. In older texts the phase is sometimes called the amplitude.
Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e.g. Dyson series). Simple and exact solutions are precluded by the fact that the system's constituents either influence each other in a complex way, or depend on temporally varying external forces.
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative series where each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics (QED) at the second order the difference from experimental data is in the order of 10−10. This close agreement holds because the coupling constant of QED is much less than 1. Notice that in this article Planck units are used, so that ħ = 1.
For conservative systems, the goal of quantum mechanics in non-perturbative regimes is to find the eigenvalues and eigenvectors of a Hamiltonian of the form
where is separable in some coordinate system, is non-separable in the coordinate system in which is separated, and is a parameter which cannot be considered small. Physicists have historically approached problems of this nature by trying to find the coordinate system in which the non-separable Hamiltonian is smallest and then treating the non-separable Hamiltonian as a perturbation.
Finding constants of motion so that this separation can be performed can be a difficult (sometimes impossible) analytical task. Solving the classical problem can give valuable insight into solving the quantum problem. If there are regular classical solutions of the same Hamiltonian, then there are (at least) approximate constants of motion, and by solving the classical problem, we gain clues how to find them.
Other approaches have been developed in recent years. One is to express the Hamiltonian in different coordinate systems in different regions of space, minimizing the non-separable part of the Hamiltonian in each region. Wavefunctions are obtained in these regions, and eigenvalues are obtained by matching boundary conditions.
Another approach is numerical matrix diagonalization. If the Hamiltonian matrix is computed in any complete basis, eigenvalues and eigenvectors are obtained by diagonalizing the matrix. However, all complete basis sets are infinite, and we need to truncate the basis and still obtain accurate results. These techniques boil down to choosing a truncated basis from which accurate wavefunctions can be constructed. The computational time required to diagonalize a matrix scales as , where is the dimension of the matrix, so it is important to choose the smallest basis possible from which the relevant wavefunctions can be constructed. It is also convenient to choose a basis in which the matrix is sparse and/or the matrix elements are given by simple algebraic expressions because computing matrix elements can also be a computational burden.
A given Hamiltonian shares the same constants of motion for both classical and quantum dynamics. Quantum systems can also have additional quantum numbers corresponding to discrete symmetries (such as parity conservation from reflection symmetry). However, if we merely find quantum solutions of a Hamiltonian which is not approachable by perturbation theory, we may learn a great deal about quantum solutions, but we have learned little about quantum chaos. Nevertheless, learning how to solve such quantum problems is an important part of answering the question of quantum chaos.
Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems. Random matrix theory was developed in an attempt to characterize spectra of complex nuclei. The remarkable result is that the statistical properties of many systems with unknown Hamiltonians can be predicted using random matrices of the proper symmetry class. Furthermore, random matrix theory also correctly predicts statistical properties of the eigenvalues of many chaotic systems with known Hamiltonians. This makes it useful as a tool for characterizing spectra which require large numerical efforts to compute.
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.
A number of statistical measures are available for quantifying spectral features in a simple way. It is of great interest whether or not there are universal statistical behaviors of classically chaotic systems. The statistical tests mentioned here are universal, at least to systems with few degrees of freedom (Berry and Taborhave put forward strong arguments for a Poisson distribution in the case of regular motion and Heusler et al. present a semiclassical explanation of the so-called Bohigas–Giannoni–Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics). The nearest-neighbor distribution (NND) of energy levels is relatively simple to interpret and it has been widely used to describe quantum chaos.
Qualitative observations of level repulsions can be quantified and related to the classical dynamics using the NND, which is believed to be an important signature of classical dynamics in quantum systems. It is thought that regular classical dynamics is manifested by a Poisson distribution of energy levels:
In addition, systems which display chaotic classical motion are expected to be characterized by the statistics of random matrix eigenvalue ensembles. For systems invariant under time reversal, the energy-level statistics of a number of chaotic systems have been shown to be in good agreement with the predictions of the Gaussian orthogonal ensemble (GOE) of random matrices, and it has been suggested that this phenomenon is generic for all chaotic systems with this symmetry. If the normalized spacing between two energy levels is , the normalized distribution of spacings is well approximated by
Many Hamiltonian systems which are classically integrable (non-chaotic) have been found to have quantum solutions that yield nearest neighbor distributions which follow the Poisson distributions. Similarly, many systems which exhibit classical chaos have been found with quantum solutions yielding a Wigner-Dyson distribution, thus supporting the ideas above. One notable exception is diamagnetic lithium which, though exhibiting classical chaos, demonstrates Wigner (chaotic) statistics for the even-parity energy levels and nearly Poisson (regular) statistics for the odd-parity energy level distribution.
Periodic-orbit theory gives a recipe for computing spectra from the periodic orbits of a system. In contrast to the Einstein–Brillouin–Keller method of action quantization, which applies only to integrable or near-integrable systems and computes individual eigenvalues from each trajectory, periodic-orbit theory is applicable to both integrable and non-integrable systems and asserts that each periodic orbit produces a sinusoidal fluctuation in the density of states.
The principal result of this development is an expression for the density of states which is the trace of the semiclassical Green's function and is given by the Gutzwiller trace formula:
Recently there was a generalization of this formula for arbitrary matrix Hamiltonians that involves a Berry phase-like term stemming from spin or other internal degrees of freedom. distinguishes the primitive periodic orbits: the shortest period orbits of a given set of initial conditions. is the period of the primitive periodic orbit and is its classical action. Each primitive orbit retraces itself, leading to a new orbit with action and a period which is an integral multiple of the primitive period. Hence, every repetition of a periodic orbit is another periodic orbit. These repetitions are separately classified by the intermediate sum over the indices . is the orbit's Maslov index. The amplitude factor, , represents the square root of the density of neighboring orbits. Neighboring trajectories of an unstable periodic orbit diverge exponentially in time from the periodic orbit. The quantity characterizes the instability of the orbit. A stable orbit moves on a torus in phase space, and neighboring trajectories wind around it. For stable orbits, becomes , where is the winding number of the periodic orbit. , where is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a difficulty because at a classical bifurcation. This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo-absorption spectrum.The index
Using the trace formula to compute a spectrum requires summing over all of the periodic orbits of a system. This presents several difficulties for chaotic systems: 1) The number of periodic orbits proliferates exponentially as a function of action. 2) There are an infinite number of periodic orbits, and the convergence properties of periodic-orbit theory are unknown. This difficulty is also present when applying periodic-orbit theory to regular systems. 3) Long-period orbits are difficult to compute because most trajectories are unstable and sensitive to roundoff errors and details of the numerical integration.
Gutzwiller applied the trace formula to approach the anisotropic Kepler problem (a single particle in a potential with an anisotropic mass tensor) semiclassically. He found agreement with quantum computations for low lying (up to ) states for small anisotropies by using only a small set of easily computed periodic orbits, but the agreement was poor for large anisotropies.
The figures above use an inverted approach to testing periodic-orbit theory. The trace formula asserts that each periodic orbit contributes a sinusoidal term to the spectrum. Rather than dealing with the computational difficulties surrounding long-period orbits to try to find the density of states (energy levels), one can use standard quantum mechanical perturbation theory to compute eigenvalues (energy levels) and use the Fourier transform to look for the periodic modulations of the spectrum which are the signature of periodic orbits. Interpreting the spectrum then amounts to finding the orbits which correspond to peaks in the Fourier transform.
Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it. In step 4 it restricts you to orbits where initial and final momentum are the same i.e. periodic orbits. Often it is nice to choose a coordinate system parallel to the direction of movement, as it is done in many books.
Closed-orbit theory was developed by J.B. Delos, M.L. Du, J. Gao, and J. Shaw. It is similar to periodic-orbit theory, except that closed-orbit theory is applicable only to atomic and molecular spectra and yields the oscillator strength density (observable photo-absorption spectrum) from a specified initial state whereas periodic-orbit theory yields the density of states.
Only orbits that begin and end at the nucleus are important in closed-orbit theory. Physically, these are associated with the outgoing waves that are generated when a tightly bound electron is excited to a high-lying state. For Rydberg atoms and molecules, every orbit which is closed at the nucleus is also a periodic orbit whose period is equal to either the closure time or twice the closure time.
According to closed-orbit theory, the average oscillator strength density at constant is given by a smooth background plus an oscillatory sum of the form
is a phase that depends on the Maslov index and other details of the orbits. is the recurrence amplitude of a closed orbit for a given initial state (labeled ). It contains information about the stability of the orbit, its initial and final directions, and the matrix element of the dipole operator between the initial state and a zero-energy Coulomb wave. For scaling systems such as Rydberg atoms in strong fields, the Fourier transform of an oscillator strength spectrum computed at fixed as a function of is called a recurrence spectrum, because it gives peaks which correspond to the scaled action of closed orbits and whose heights correspond to .
Closed-orbit theory has found broad agreement with a number of chaotic systems, including diamagnetic hydrogen, hydrogen in parallel electric and magnetic fields, diamagnetic lithium, lithium in an electric field, the ion in crossed and parallel electric and magnetic fields, barium in an electric field, and helium in an electric field.
For the case of one-dimensional system with the boundary condition the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by here is the density of states and V(x) is the classical potential of the particle, the half derivative of the inverse of the potential is related to the density of states as in the Wu-Sprung potential
The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the study of eigenfunctions (Quantum ergodicity, scars) of various chaotic Hamiltonian .
Further studies concern the parametric () dependence of the Hamiltonian, as reflected in e.g. the statistics of avoided crossings, and the associated mixing as reflected in the (parametric) local density of states (LDOS). There is vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc. Special place is reserved to the study of the dynamics of quantized maps: the standard map and the kicked rotator are considered to be prototype problems.
Works are also focused in the study of driven chaotic systems, is time dependent, in particular in the adiabatic and in the linear response regimes. There is also significant effort focused on formulating ideas of quantum chaos for strongly-interacting many-body quantum systems far from semiclassical regimes.where the Hamiltonian
In 1977, Berry and Tabor made a still open "generic" mathematical conjecture which, stated roughly, is: In the "generic" case for the quantum dynamics of a geodesic flow on a compact Riemann surface, the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely integrable.
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces. Many of these structures are drawn from functional analysis, a research area within pure mathematics that was influenced in part by the needs of quantum mechanics. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the motion of atomic nuclei and electrons in a molecule can be treated separately. The approach is named after Max Born and J. Robert Oppenheimer who proposed it in 1927, in the early period of quantum mechanics. The approximation is widely used in quantum chemistry to speed up the computation of molecular wavefunctions and other properties for large molecules. There are cases where the assumption of separable motion no longer holds, which make the approximation lose validity, but is then often used as a starting point for more refined methods.
Loop quantum gravity (LQG) is a theory of quantum gravity attempting to merge quantum mechanics and general relativity, including the incorporation of the matter of the standard model into the framework established for the pure quantum gravity case. LQG competes with string theory as a candidate for quantum gravity, but unlike string theory is not a candidate for a theory of everything.
In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation diverge at a rate given by
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior.
In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation, master equation in Lindblad form, or Lindbladian is the most general type of Markovian and time-homogeneous master equation describing evolution of the density matrix ρ that preserves the laws of quantum mechanics.
Quantum numbers describe values of conserved quantities in the dynamics of a quantum system. In the case of electrons, the quantum numbers can be defined as "the sets of numerical values which give acceptable solutions to the Schrödinger wave equation for the hydrogen atom". In more general cases, quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be known with precision at the same time as the system's energy—and their corresponding eigenspaces. Together, a specification of all of the quantum numbers of a quantum system fully characterize a basis state of the system, and can in principle be measured together.
The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus. The question of how the operational process measurement affects the ontological state of the observed system is unresolved, and called the measurement problem.
In mathematics, the Hilbert–Pólya conjecture is a possible approach to the Riemann hypothesis, by means of spectral theory.
In statistical mechanics, a micro-canonical ensemble is the statistical ensemble that is used to represent the possible states of a mechanical system that has an exactly specified total energy. The system is assumed to be isolated in the sense that the system cannot exchange energy or particles with its environment, so that the energy of the system remains exactly the same as time goes on. The system's energy, composition, volume, and shape are kept the same in all possible states of the system.
The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as the semi-classical approximation to modern quantum mechanics.
In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the invariant measure of the system.
In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems and discrete systems. The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them.
In quantum mechanics, given a particular Hamiltonian and an operator with corresponding eigenvalues and eigenvectors given by , then the numbers are said to be "good quantum numbers" if every eigenvector remains an eigenvector of with the same eigenvalue as time evolves.
In physics, and especially quantum chaos, a quantum scar is a kind of quantum state with a high likelihood of existing in unstable classical periodic orbits in classically chaotic systems. The term also refers to the wave function of such a state, which is more formally defined by having an enhancement of an eigenfunction along unstable classical periodic orbits. Since the square-norm of quantum wavefunctions yield probability densities in the Copenhagen interpretation, the two notions correspond.
Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators.