Determinism, Chaos and Quantum Mechanics.(en)(30s)

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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. Hence, every repetition of a periodic orbit is another periodic orbit. Neighboring trajectories of an unstable periodic orbit diverge exponentially in time from the periodic orbit. A stable orbit moves on a torus in phase space, and neighboring trajectories wind around it. This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo- absorption spectrum.

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. This difficulty is also present when applying periodic-orbit theory to regular systems.

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.

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. Delos, M. Du, J.

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Gao, and J. 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.

Reality Club Discussion

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. 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.

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. Recent [ when?

There is also significant effort focused on formulating ideas of quantum chaos for strongly-interacting many-body quantum systems far from semiclassical regimes. In , 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.

From Wikipedia, the free encyclopedia. Branch of physics seeking to explain chaotic dynamical systems in terms of quantum theory. This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts , without removing the technical details. October Learn how and when to remove this template message. Classical mechanics Old quantum theory Bra—ket notation Hamiltonian Interference.

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Quantum annealing Quantum chaos Quantum computing Density matrix Quantum field theory Fractional quantum mechanics Quantum gravity Quantum information science Quantum machine learning Perturbation theory quantum mechanics Relativistic quantum mechanics Scattering theory Spontaneous parametric down-conversion Quantum statistical mechanics. Berry and M.

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Jul 8, Free will has nothing to do with quantum mechanics. We are Modern physics has altered the data a bit, and the ensuing confusion requires. Determinism, Chaos and Quantum Mechanics. Jean Bricmont Institut de Physique Th´eorique Universit´e Catholique de Louvain, Louvain-la-Neuve, BELGIUM.

Tabor, Proc. Altland, P. Braun, and F. Haake, , Phys. Physical Review B. Bibcode : PhRvB.. Garbaczewski and R. Olkiewicz Springer, Bibcode : EL Jan Notices of the AMS. Chaos theory. For these philosophers, there is a simple consequence: determinism is a false doctrine. As with the Humean view, this does not mean that concerns about human free action are automatically resolved; instead, they must be addressed afresh in the light of whatever account of physical nature without laws is put forward. We can now put our—still vague—pieces together. Determinism requires a world that a has a well-defined state or description, at any given time, and b laws of nature that are true at all places and times.

If we have all these, then if a and b together logically entail the state of the world at all other times or, at least, all times later than that given in a , the world is deterministic.

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How could we ever decide whether our world is deterministic or not? Given that some philosophers and some physicists have held firm views—with many prominent examples on each side—one would think that it should be at least a clearly decidable question. Unfortunately, even this much is not clear, and the epistemology of determinism turns out to be a thorny and multi-faceted issue.

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As we saw above, for determinism to be true there have to be some laws of nature. Most philosophers and scientists since the 17 th century have indeed thought that there are. But in the face of more recent skepticism, how can it be proven that there are?

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And if this hurdle can be overcome, don't we have to know, with certainty, precisely what the laws of our world are , in order to tackle the question of determinism's truth or falsity? The first hurdle can perhaps be overcome by a combination of metaphysical argument and appeal to knowledge we already have of the physical world.

Philosophers are currently pursuing this issue actively, in large part due to the efforts of the anti-laws minority. The debate has been most recently framed by Cartwright in The Dappled World Cartwright in terms psychologically advantageous to her anti-laws cause.

Those who believe in the existence of traditional, universal laws of nature are fundamentalists ; those who disbelieve are pluralists.