  Excitons, as bound states of electrons and holes, embody the solid state
analogue of the hydrogen atom, whose quantum spectrum is explained within a
classical framework by the Bohr-Sommerfeld atomic model. In a first
hydrogenlike approximation the spectra of excitons are also well described by a
Rydberg series, however, due to the surrounding crystal environment deviations
from this series can be observed. A theoretical treatment of excitons in
cuprous oxide needs to include the band structure of the crystal, leading to a
prominent fine-structure splitting in the quantum spectra. This is achieved by
introducing additional spin degrees of freedom into the system, making the
existence and meaningfulness of classical exciton orbits in the physical system
a non-trivial question. Recently, we have uncovered the contributions of
periodic exciton orbits directly in the quantum mechanical recurrence spectra
of cuprous oxide [J. Ertl et al., Phys. Rev. Lett. 129, 067401 (2022)] by
application of a scaling technique and fixing the energy of the classical
dynamics to a value corresponding to a principle quantum number $n=5$ in the
hydrogenlike case. Here, we present a comprehensive derivation of the classical
and semiclassical theory of excitons in cuprous oxide. In particular, we
investigate the energy dependence of the exciton dynamics. Both the
semiclassical and quantum mechanical recurrence spectra exhibit stronger
deviations from the hydrogenlike behavior with decreasing energy, which is
related to a growing influence of the spin-orbit coupling and thus a higher
velocity of the secular motion of the exciton orbits. The excellent agreement
between semiclassical and quantum mechanical exciton recurrence spectra
demonstrates the validity of the classical and semiclassical approach to
excitons in cuprous oxide.
