Again, it should be stressed, that Einsteins quantum hypothesis was just heuristics, and indeed Einstein titled one of his famous papers of his "annus mirabilis 1905" (although I think that Einsteins true "annus mirabilis" was 1915, where he finished his General Theory of Relativity) "Über einen die Erzeugung und Verwandlung des Lichts betreffenden heuristischen Gesichtspunkt" (which I'd freely translate as "About an heuristic aspect concerning the production and transformation of light"). It is together with the Bohr-Sommerfeld model of atoms and de Broglie's first version of wave-particle dualism, so to say, a precursor theory which is outdated for more than 90 years now by modern quantum theory. Particularly photons have very little in common with a classical-particle picture, and as any quantum in the relativistic realm, cannot be understood in terms of Schrödinger single-particle wave functions as can be to quite an extent for non-relativistic particles in terms of Schrödinger wave mechanics.
The reason is that in the relativistic realm interactions between quanta alway can lead to the production and destruction of quanta, and there are no conserved positive definite particle numbers. What's conserved are similar quantum numbers like electric charge (which allows to create particle-antiparticle pairs without changing the total charge) or net-baryon number (which allows to create baryon-antibaryon pairs without changing the total baryon number) etc. That's why you need a formalism that is able to describe the production and destruction of particles when it comes to relativistic collision energies (i.e., energies that are larger than the threshold of particle production, and since photons are massless there's no such threshold at all, i.e., the acceleration of any charged particles is always accompanied by the production of photons). Such a formalism now is quantum field theory, and free fields can be decomposed in energy-momentum eigenmodes, and since there's no interaction these modes are decoupled, i.e., you can treat them as an infinite set of uncoupled harmonic oscillators. Now the harmonic oscillator can be solved in quantum mechanics by introducing the annihilation and creation operators of energy eigenmodes, and these are in the QFT precisely the annihilation and creation operators of field quanta. These annihilation and creation operators have as a very convenient side effect the property of leading to the "occupation-number" or Fock basis which automatically take into account the bosonic or fermionic nature of the quanta, i.e., it leads to a basis of properly symmetrized (bosons) or anti-symmetrized (fermions) many-body states.
The Einstein-de-Broglie rule for free fields, i.e., associating ##\hbar \omega## with the energy and ##\hbar \vec{k}## with the momentum of a single quantum of this field, survives the quantization of the field theory, because for free fields the equations of motion are linear in the fields, and thus the field operators in the non-interacting case obey the same equations of motion as the classical theory. Thus each frequency eigenmode of the quantized field has the same frequency-wavenumber dispersion relation as the classical field and thus, since the Einstein-de-Broglie rule of old quantum mechanics survives the modern way of QT, the energy-momentum relation of single non-interacting particles, i.e., ##E^2 c^2-\vec{p}^2=m^2 c^2## for a quantum with (invariant) mass ##m##. For a photon you have ##m_{\gamma}=0## in the Standard Model of elementary particles, and the empirical upper limit of the photon mass is indeed very small ##m_{\gamma}<10^{-18} \,\text{eV}##.
For more about the Standard Model, see the marvelous popular-science page by the Particle Data Group:
http://www.particleadventure.org/
