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. Semiclassical methods Periodic orbit theory 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. The index 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