Editing Quantum chaos (section)

=== Closed orbit theory ===
[[File:Cotcomp.jpg|right|thumb|Experimental recurrence spectrum (circles) is compared with the results of the closed orbit theory of John Delos and Jing Gao for lithium [[Rydberg atom]]s in an electric field.  The peaks labeled 1–5 are repetitions of the electron orbit parallel to the field going from the nucleus to the classical turning point in the uphill direction.]]
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 <math>\epsilon</math> is given by a smooth background plus an oscillatory sum of the form
: <math>
f(w) = \sum_k \sum_{n=1}^{\infty} D^{i}_{\it nk}
\sin(2\pi nw\tilde{S_k} - \phi_{\it nk}).
</math>
<math>\phi_{\it nk}</math> is a phase that depends on the Maslov index and other details of the orbits. <math>D^i_{\it nk}</math> is the recurrence amplitude of a closed orbit for a given initial state (labeled <math>i</math>).  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 <math>\epsilon</math> as a function of <math>w</math> is called a recurrence spectrum, because it gives peaks which correspond to the scaled action of closed orbits and whose heights correspond to <math>D^i_{\it nk}</math>.

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 <math>H^{-}</math> ion in crossed and parallel electric and magnetic fields, barium in an electric field, and helium in an electric field.