Quantum Ergodicity
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quantum chaos Quantum chaos is a branch of physics focused on 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 ...
, a branch of
mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, quantum ergodicity is a property of the quantization of classical mechanical systems that are
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program aired on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids, Cartoon Netwo ...
in the sense of exponential sensitivity to initial conditions. Quantum ergodicity states, roughly, that in the high-energy limit, the probability distributions associated to
energy eigenstates Energy () is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat and light. Energy is a conserved quantity—the law of conservation of energy sta ...
of a quantized
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
tend to a uniform distribution in the classical
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
. This is consistent with the intuition that the flows of ergodic systems are equidistributed in phase space. By contrast, classical completely integrable systems generally have periodic orbits in phase space, and this is exhibited in a variety of ways in the high-energy limit of the eigenstates: typically, some form of concentration occurs in the semiclassical limit \hbar \rightarrow 0. The model case of a Hamiltonian is the geodesic Hamiltonian on the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
of a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
. The quantization of the geodesic flow is given by the
fundamental solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ...
of the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
:U_t=\exp(it\sqrt) where \sqrt is the square root of the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named aft ...
. The quantum ergodicity theorem of Shnirelman 1974, Zelditch, and
Yves Colin de Verdière Yves Colin de Verdière is a French mathematician. Life He studied at the École Normale Supérieure in Paris in the late 1960s, obtained his Ph.D. in 1973, and then spent the bulk of his working life as faculty at Joseph Fourier University in ...
states that a compact Riemannian manifold whose
unit tangent bundle In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (''M'', ''g''), denoted by T1''M'', UT(''M''), UT''M'', or S''M'' is the unit sphere bundle for the tangent bundle T(''M''). It is a fiber bundle over ''M'' whose fiber at eac ...
is ergodic under the geodesic flow is also ergodic in the sense that the probability density associated to the ''n''th
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Laplacian tends weakly to the uniform distribution on the unit cotangent bundle as ''n'' → ∞ in a subset of the natural numbers of
natural density In number theory, natural density, also referred to as asymptotic density or arithmetic density, is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desi ...
equal to one. Quantum ergodicity can be formulated as a non-commutative analogue of the classical ergodicity ( T. Sunada). Since a classically chaotic system is also ergodic, almost all of its trajectories eventually explore uniformly the entire accessible phase space. Thus, when translating the concept of ergodicity to the quantum realm, it is natural to assume that the eigenstates of the quantum chaotic system would fill the quantum phase space evenly (up to random fluctuations) in the semiclassical limit \hbar \rightarrow 0. The quantum ergodicity theorems of Shnirelman, Zelditch, and
Yves Colin de Verdière Yves Colin de Verdière is a French mathematician. Life He studied at the École Normale Supérieure in Paris in the late 1960s, obtained his Ph.D. in 1973, and then spent the bulk of his working life as faculty at Joseph Fourier University in ...
proves that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. However, the quantum ergodicity theorem leaves open the possibility of eigenfunctions become sparse with serious holes as \hbar \rightarrow 0, leaving large but not macroscopic gaps on the energy manifolds in the phase space. In particular, the theorem allows the existence of a subset of macroscopically nonergodic states which on the other hand must approach zero measure, i.e., the contribution of this set goes towards zero percent of all eigenstates when \hbar \rightarrow 0. For example, the theorem do not exclude quantum scarring, as the phase space volume of the scars also gradually vanishes in this limit. A quantum eigenstate is scarred by periodic orbit if its probability density is on the classical invariant manifolds near and all along that periodic orbit is systematically enhanced above the classical, statistically expected density along that orbit. In a simplified manner, a quantum scar refers to an eigenstate of whose probability density is enhanced in the neighborhood of a classical periodic orbit when the corresponding classical system is chaotic. In conventional scarring, the responsive periodic orbit is unstable. The instability is a decisive point that separates quantum scars from a more trivial finding that the probability density is enhanced near stable periodic orbits due to the Bohr's
correspondence principle In physics, a correspondence principle is any one of several premises or assertions about the relationship between classical and quantum mechanics. The physicist Niels Bohr coined the term in 1920 during the early development of quantum theory; ...
. The latter can be viewed as a purely classical phenomenon, whereas in the former quantum interference is important. On the other hand, in the perturbation-induced quantum scarring, some of the high-energy eigenstates of a locally perturbed quantum dot contain scars of short periodic orbits of the corresponding unperturbed system. Even though similar in appearance to ordinary quantum scars, these scars have a fundamentally different origin., In this type of scarring, there are no periodic orbits in the perturbed classical counterpart or they are too unstable to cause a scar in a conventional sense. Conventional and perturbation-induced scars are both a striking visual example of classical-quantum correspondence and of a quantum suppression of chaos (see the figure). In particular, scars are a significant correction to the assumption that the corresponding eigenstates of a classically chaotic Hamiltonian are only featureless and random. In some sense, scars can be considered as an eigenstate counterpart to the quantum ergodicity theorem of how short periodic orbits provide corrections to the universal
random matrix In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the ...
theory eigenvalue statistics.


See also

* Eigenstate thermalization hypothesis *
Ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., tha ...
*
Quantum chaos Quantum chaos is a branch of physics focused on 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 ...
* Scar (physics)


External links


Shnirelman theorem, Scholarpedia article


References

* * * {{Citation , last=Sunada, first= T, chapter=Quantum ergodicity , title=Trend in Mathematics , publisher=Birkhauser Verlag, Basel, year=1997 , pages=175–196 Modular forms Chaos theory Ergodic theory Quantum mechanics Quantum chaos theory