Semiclassical interpretation of eigenvectors for excited atoms in external fields

Eigenvectors for an electron in an atom in parallel electric and magnetic fields are calculated, and a semiclassical interpretation of their behavior is obtained. Eigenvectors can in this case be regarded as ‘‘wave functions in angular momentum space.’’ The matrix equation defining the eigenvectors is written as a difference equation, and then converted to a pseudodifferential equation; a systematic procedure is then used to construct a semiclassical approximation. It is found that the same classical Hamiltonian that has been previously used to calculate semiclassical eigenvalues provides a WKB-type representation of the eigenvectors. The development sheds new light on action-angle formulations of quantum mechanics and on semiclassical approximations in action-angle variables.