The time-independent Shrödinger equation (TISE) is
$$\dfrac{-\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi,$$
where $m$ is the mass of the particle.
I just had a thought: If $m$ is the particle's mass, then the TISE is invalid for photons, since they're massless! But I know that the Shrödinger equation is used to model photons. So how can this be?
The Schrödinger equation can not be used to model photons. In order to understand why, I think it is useful to use the effective field theory machinery here, since Shcrödinger-like equations can be obtained as the "equation of motion" of Non-Relativistic Quantum Electrodynamics theories (NRQED). In these theories, in the reference frame where the massive particle is at rest, the leading order operator is of the form (in the abscence of EM field):
$$i\dfrac{\partial}{\partial t}+\dfrac{\hbar^2}{2m}\nabla^2$$
Which is the Kinetic part of the Schrödinger equation. You can also add interactions. I recommend you look at any recent review of NRQED at the arxiv, such as this one.
