We all have studied in introduction to quantum mechanics about Compton Effect. In all the books I have read, it says that classical theory can't explain the shift in wavelength because the incident EM wave will oscillate the electron at the frequency of light, and the oscillating electron will emit radiation of the same frequency. This is all well and good, but while reading Quantum Physics[Berkeley Series],the author mentioned something among the lines that ( not exact sentence) , the em waves oscillate electrons and that in turn produces the light of same frequency, also some loosely bound electrons are ejected from the atom which radiate the light of the slightly different frequency.
Is this explanation correct, if so what was the problem in mathematically describing this theory? Also , if this is the case, how is the ejected electron different from the ones in photoelectric effect.