What equation describes the wavefunction of a single photon? The Schrödinger equation describes the quantum mechanics of a single massive non-relativistic particle. The Dirac equation governs a single massive relativistic spin-½ particle. The photon is a massless, relativistic spin-1 particle.
What is the equivalent equation giving the quantum mechanics of a single photon?
 A: The general concept of quantum mechanics is that particles are waves. One of the hand-waveing "derivations" of quantum mechanics assumes that the phase of particles behaves in the same way as the phase of light $\exp( i \vec{k}\cdot \vec{x} - iE t / \hbar)$ (see Feynman Lectues on Physics, Volume 3, Chapter 7-2).
For light that is monochromatic (or almost monochromatic), just take Maxwell Equations plus add the assumption that one photon can't be partially absorbed. Most of the time it suffices to use the paraxial approximation, or even - plane wave approximation. It works for standard quantum mechanics setups like
Elitzur–Vaidman bomb-tester.
For nonmonochronatic light it's much more complicated. More on the nature of quantum mechanics of one photon:
Iwo Bialynicki-Birula, On the Wave Function of the Photon, Acta Physica Polonica 86, 97-116 (1994).
A: There is no quantum mechanics of a photon, only a quantum field theory of electromagnetic radiation. The reason is that photons are never non-relativistic and they can be freely emitted and absorbed, hence no photon number conservation. 
Still, there exists a direction of research where people try to reinterpret certain quantities of electromagnetic field in terms of the photon wave function, see for example this paper.
A: A single photon is described quantum mechanically by the Maxwell equations, where the solutions are taken to be complex.  The Maxwell equations can be written in the form of the matrix Dirac equation, where the Pauli two-component matrices, corresponding to spin 1/2 electrons, are replaced by analogous three-component matrices, corresponding to spin 1 photons.  Since the Dirac equation and corresponding Maxwell equation are fully relativistic, there is no problem with the mass of the photon being zero, as there would be for a Schroedinger-like equation.  See http://www.nist.gov/pml/div684/fcdc/upload/preprint.pdf.
A: According to Wigner's analysis, the single photon Hilbert space is spanned by a basis parameterized by energy-momenta on the forward light cone boundary, and a helicity of $\pm 1$. 
However, a manifestly Lorentz covariant description in position space has to include a fictitious longitudinal photon with a helicity of 0. This degree of freedom is pure gauge, and decouples. Interestingly enough, the state norm is now positive semidefinite, instead of positive definite, with the transverse modes having positive norm and the longitudinal ones having zero norm.
A: There are several different waves associated with a photon. In QED the photon is associated with a classical solution of the (4-)vector potential. The vector potential contains features that are not physical, as a change of gauge is not reflected in any change of physical properties. Thus its role as a wave function might be somewhat questionable.But still, there must be a wave which explains the well known interference and diffraction patterns.
When we see a screen illuminated by laser light that have passed through a double slit, our eyes receives photons scattered by the atoms on the surface of the screen. Atoms absorb and emit photons as quantum electric dipole antennas. This implies that the atoms are sensitive to the electric field. From the vector field associated with the photon an electric field can be calculated. This field is gauge independent thus a physical field. This field is a solution to Maxwell's equations and will describes the usual interference and diffraction patterns. 
A: My answer is more of comment on other correct answers: you cannot build a delta-function for the photon in 3D becase the longitudinal component of a massless vector field is missing. But that does not mean there is no useful and meaningful concept of a wave function in the single-photon sector. This is just a peculiar fact about free electromagnetic field, you basically cannot localize light to a region smaller that the characteristic wave-length. Maxwell equations for the source-less (solenoidal) component of the vector potential field $\bf{A}$ play the role of the Schrodinger equation.
I recommend the book of Rodney Loudon "The Quantum Theory of Light" as good resource to really understand the quantum level of desription for light.
A: There is a slight confusion in this question. In quantum field theory, the Dirac equation and the Schrödinger equation have very different roles. The Dirac equation is an equation for the field, which is not a particle. The time evolution of a particle, ie, a quantum state, is always given by the Schrödinger equation. The hamiltonian for this time evolution is written in terms of fields which obey a certain equation themselves. So, the proper answer is: Schrödinger equation with a hamiltonian given in terms of a massless vector field whose equation is nothing else but Maxwell's equation.
A: The maxwell equations, just like in classical electrodynamics. You'll need to use quantum field theory to work with them though.
http://en.wikipedia.org/wiki/Relativistic_wave_equations
http://en.wikipedia.org/wiki/Quantum_electrodynamics
A: While the answers above are great, I felt it was lacking what The question asked regarding an equation analogous to the Schrodinger (or Dirac) equation.
There is a quantity called the Riemann-Silberstein vector (https://en.wikipedia.org/wiki/Riemann%E2%80%93Silberstein_vector#Photon_wave_function), first used by the famous Bernhardt Riemann to demonstrate a concise formulation of Maxwell's equations.
This “vector” has the form:
$$\vec{F}=\vec{E}+ic\vec{B}$$
A quick search online, demonstrates that classical electrodynamics written in this form can be quite useful in solving problems.
In the quantum realm, a quantity analogous to the wavefunction can be written for a single photon. Such a quantity has the form:
$$i\hbar\partial_{t}\vec{F}=c\left(\vec{S}\cdot\frac{\hbar}{i}\vec{\nabla}\right)\vec{F}$$ Which can be written simply in the form:
$$i\hbar\partial_{t}\vec{F}=c\left(\vec{S}\cdot\hat{P}\right)\vec{F}$$
This can be a useful quantity for examining the properties of a single photon. Start with the Wikipedia page, it's actually quite an interesting and useful quantity.
A: There is a nice way to represent Maxwell's equations using 2-component spinor formalism. Final expression is indeed a wave equation but it is best to interpret this as a semi-classical result.
One can decompose the Maxwell tensor into anti self dual and self dual parts, which is represented in spinorial form:
$$F_{ab} \leftrightarrow F_{ABA'B'}=\phi_{AB}\epsilon_{A'B'}+\bar{\phi}_{A'B'}\epsilon_{AB}$$
The small latin indices are the space-time indices while the capital primed and un-primed are the spinor indices. Also note that {a} <---> {AA'} , unprimed indices like A takes value either 0 or 1, while it is 0' or 1' for primed indices (like A')
The final expression for Maxwell's equation (in cgs unit) looks like :
$$\nabla^A_{B'}\phi_{AB}=-2\pi J_{BB'}$$ where J is the current density. In flat space-time we have
$$\nabla_{AB'}=\frac{\partial}{\partial x^{AB'}}$$ In Minkowski null tetrad (also referred to as Lightcone gauge in other literatures) we have $x^{00'}=\frac{t+z}{\sqrt{2}}$, $x^{01'}=\frac{x+iy}{\sqrt{2}}$ and so on..
The source free Maxwell field $\phi_{AB}$ (where $\nabla^A_{B'}\phi_{AB}=0$) when expressed in terms of Twistor functions, appears to have a non-local property which is quite similar to non-local nature of wave functions in quantum mechanics. In twistor literature, this relation: $$\nabla^A_{B'}\phi_{AB}=0$$ is often referred to as wave equation for source free mass-less spin 1 field, where $\phi_{AB}$ is the "wave function".
