Wigner transforms, eigenfunctions

The concentration of the Wigner transforms of the eigenfunctions on the quantized tori provides the counterpart in classically integrable systems to the scarring phenomena associated with unstable periodic orbits. 

Concerning the Poincare surface of section, it should be noticed that a sort of quantum surface of section can be constructed by intersection of the Wigner or Husimi transform of the eigenfunctions expressed in the quantum action-angle variables of the effective Hamiltonian, which can provide a comparison with the classical Poincare surfaces of section (e.g., in acetylene). 

Derivation of the Wigner-Eckart theorem (WET) will be based on the transformation properties of the energy eigenfunctions expressed by... 

Equation (3.32a) implies normalization, and Eq. (3.32b) contains the essential probabilistic interpretation of the projections onto the momenta or coordinates. The last condition, a natural consequence of the definitions of the density matrix and the Wigner-Weyl transform, explicitly eliminates the singular distributions allowed in Eq. (3.31d). That is, although the completeness of the quantum PijXp, q) basis permits the construction of 8 function distributions, they make, unlike classical mechanics, no natural appearance in quantum mechanics wherein eigenfunctions of LQ are square integrable and such singular distributions are explicitly excluded in Eq. (3.32d). 

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