page_title: Quantum chaos
section_title: Quantum mechanics in non-perturbative regimes
hierarchical_section_title: Quantum chaos / Quantum mechanics in non-perturbative regimes
caption_attribution_description: Energy level spectra of lithium in an electric field
context_page_description: 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. 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?
  In seeking to address the basic question of quantum chaos, several approaches have been employed:
  Development of methods for solving quantum problems where the perturbation cannot be considered small in perturbation theory and where quantum numbers are large.
  Correlating statistical descriptions of eigenvalues with the classical behavior of the same Hamiltonian.
context_section_description: 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.
caption: Computed chaotic Rydberg atom energy level spectra of lithium in an electric field near n=15.  Note that energy levels cannot cross due to the ionic core (and resulting quantum defect) breaking symmetries of dynamical motion.[4]
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