A Quora Question Asks: Is it true that there is no Schrodinger equation for light because the Schrodingers’ equation is only for massive particles, and that only Maxwell’s equations can be used for light?
My answer is that it depends on what you mean by the Schrödinger Equation.
The first time many of us see that name is in reference to the position space wave equation for a single massive particle in non-relativistic quantum mechanics (and that is indeed historically appropriate, but is not the only equation that is correctly attributed to Schrödinger). In terms of this usage, the Schrödinger Equation clearly does not apply to light because the behaviour of light is not Galilean-covariant (and, unlike particles, does not even have a useful Galilean-covariant approximation).
But the name “Schrödinger Equation” is also used for the equation that governs the time evolution of any quantum system (which is basically just a statement of the fact that any unitary representation of the one dimensional translation group has a self adjoint generator – which, in the case of time translations, we identify as corresponding to the total energy of the corresponding spacelike time slice). And in that sense, as noted by Mark John Fernee, (with some further elaboration by Diógenes Figueroa) it certainly does apply to the quantized electromagnetic field and so to light.
There is also a Schrödinger Equation in this second sense for the Quantum Field Theory of electrons and positrons; however the relativistic wave equation for a single free electron is not called a Schrödinger Equation but rather is known as the Dirac Equation instead.
For the case of a single relativistic spinless particle, there is no wave equation that is first order in time, and the best we can do is the second order Klein-Gordon Equation; but there is nonetheless a Schrödinger Equation in the second sense for the corresponding scalar quantum field theory.
There is a sense in which Maxwell’s Equations play a similar role to the Dirac and Klein-Gordon equations (and the first interpretation of Schrödinger’s) as the wave equation for a single photon, but the use of Electric and Magnetic fields to extract a probability density is complicated by having to take account of gauge invariance and polarization state.