Ray splitting is the phenomenon whereby a ray incident on a boundary splits into more than one ray traveling away from the boundary. The most common example of this is the situation, originally considered by Snell in 1621, in which an incident light ray splits into reflected and transmitted rays at a discontinuity in refractive index. This paper seeks to extend techniques and results from quantum chaos to short wavelength problems in which ray splitting surfaces are present. These extensions are tested using a simple model problem for the Schr\"odinger equation in two dimensions with a finite step potential discontinuity. Numerical solutions for the energy spectrum and eigenfunctions in this system are then compared with predictions based on quasiclassical theoretical results suitably extended to include ray splitting. Among the topics treated are the ray orbits for our problem, energy level statistics, scars, trace formulas, the quasiclassical transfer operator technique, and the effect of lateral waves. It is found that these extensions of quantum chaos are very effective for treating problems with ray splitting.