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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?

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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.

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    $\begingroup$ You can also say that the wavefunction of a photon is defined as long as the photon is not emitted or absorbed. The wavefunction of a single photons is used in single-photon interferometry, for example. In a sense, it is not much different from the electron, where the wave-function start to be problematic when electrons start to be created or annihilated... $\endgroup$ Commented Nov 17, 2010 at 10:19
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    $\begingroup$ I agree. For the electrons there is a possibility to slow them down to non-relativistic speeds, but there is no such possibility for photons. I would also add that there is an interesting discussion about photons and electrons in the Peierls's book "Surprises in theoretical physics". $\endgroup$ Commented Nov 18, 2010 at 21:46
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    $\begingroup$ Igor, I can't reconcile your wording with Frédéric's comment. Yes, there is no possibility for photons to slow down relativistically, but so what? Unless I misunderstand, there is still a spatial wavefunction (complexly valued over R^3) for the photon which obeys a relativistic Schrodinger equation. Yes, we have to assume the photon is not emitted or absorbed, but the same is true with electrons! The description of the latter in terms of a spatial wavefunction also breaks down when they are emitted or absorbed. $\endgroup$ Commented Jun 3, 2013 at 20:36
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    $\begingroup$ You should google W.E. Lamb's anti-photon paper. $\endgroup$
    – Jan Bos
    Commented Sep 29, 2016 at 13:52
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    $\begingroup$ The field does still (in the Schrodinger picture) have a state which is a vector in a Hilbert space that evolves according to the Schrodinger equation, though. It's just that said state isn't usefully interpreted in terms of number-conserved "photons". $\endgroup$
    – AGML
    Commented May 29, 2017 at 22:40
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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.

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  • $\begingroup$ The Dirac equation describes an electron. The electron is a particle. The Dirac equation contains the energy operator, which is a time derivative. Why should it not describe the time evolution of a particle? I don't get your seperation between "field" and "particle". Aren't particles always just quantized excitations of a field? Fermion field or boson field? $\endgroup$
    – Leviathan
    Commented Dec 10, 2022 at 1:02
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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.

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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

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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).

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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.

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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.

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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.

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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.

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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".

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The generalization of the Dirac equation to the cases of other than spin 1/2 is the Duffin–Kemmer-Petiau Equation. Strictly speaking, this applies to systems with non-zero rest-mass $m$, but it can be directly adapted - as is - by setting $m = 0$, to handle light-speed particles, just as it can for the Dirac equation. I don't believe there will be any complications or road-blocks in doing so.

It might be possible to mash it up with Peter Mohr's Solutions Of The Maxwell Equations And Photon Wave Functions (National Institute Of Standards And Technology) - but to handle gauge-fixing, it will require adding in an 11th component, instead of the 10 that normally go with the Kemmer equation for mass non-zero spin 1. The starting point is Mohr's $$∇·𝐄 = \frac{ρ}{ε_0},\label{1}\tag{1}$$ $$∇×𝐁 - \frac{1}{c^2}\frac{∂𝐄}{∂t} = μ_0𝐉,\label{2}\tag{2}$$ $$∇×𝐄 + \frac{∂𝐁}{∂t} = 𝟬,\label{3}\tag{3}$$ $$∇·𝐁 = 0,\label{4}\tag{4}$$ where $c = 1/\sqrt{ε_0μ_0}$.

Delve deeper into the potentials to replace ($\ref{3}$) and ($\ref{4}$) by $$-∇φ - \frac{∂𝐀}{∂t} = 𝐄,\tag{3A}$$ $$∇×𝐀 = 𝐁.\tag{4A}$$ Write down an extra equation for the Lorenz term $$∇·𝐀 + \frac{1}{c^2}\frac{∂φ}{∂t} = b.\tag{0A}\label{0A}$$ Modify ($\ref{1}$) and ($\ref{2}$) by off-setting the Lorenz term $$∇·𝐄 + \frac{∂b}{∂t} = \frac{ρ}{ε_0},\tag{1A}\label{1A}$$ $$∇×𝐁 - \frac{1}{c^2}\frac{∂𝐄}{∂t} - ∇b = μ_0𝐉.\tag{2A}\label{2A}$$ Then, try to apply the remainder of Mohr's paper to these equations, instead. The matrix representation is $11×11$, instead of the usual $10×10$ representation for the massive spin 1 case.

It's still possible to add in a mass term, and make the equations Proca equations. This requires further modification of ($\ref{1A}$) and ($\ref{2A}$) to $$∇·𝐄 + \frac{∂b}{∂t} = \frac{ρ}{ε_0} - \frac{φ}{λ^2},\tag{1B}\label{1B}$$ $$∇×𝐁 - \frac{1}{c^2}\frac{∂𝐄}{∂t} - ∇b = μ_0𝐉 - \frac{𝐀}{λ^2},\tag{2B}\label{2B}$$ corresponding to a boson of mass $m = ħ/(λc) > 0$. In both cases, the gauge needs to be fixed by applying the Lorenz condition $b = 0$, or whatever refinement to it is required for quantization. For the case $m > 0$, ($\ref{0A}$) can be removed, and $b$ can be removed from ($\ref{1B}$) and ($\ref{2B}$), because $b = 0$ will already follow as a consquence. The inclusion of $b$ is to make it easier to handle the $m = 0$ case.

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