Wavefunction of a photon

Question

Does anyone have an explicit closed-form expression for the wavefunction of a single photon from a multipolar source propagating through free space? Any basis is acceptable as long as it is a single photon state.

A reference would also be appreciated, but not essential.

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A possible duplicate has been suggested: Does a photon have a wave function or not?

But this question primarily concerns the existence of the wavefunction and is not what I am looking for. None of the answers provide an explicit expression for the wave function, and neither the question nor the answers discuss a multipole source. The multipole source, in particular, is central to my question.

Stephen Blake · Accepted Answer

I think you are asking for the wavefunction of a single photon in a state of sharp total angular momentum $j$. This is,
begin{equation}
|jmklambdarangle=frac{(2j+1)}{4pi}intsin{beta}dalpha dbeta {D^{j}(alpha,beta,0)^{m}_{ lambda}}^{*}|vec{k},lambdarangle
end{equation}
In this equation the 3-momentum of the photon is the vector $vec{k}$. The magnitude of $vec{k}$ is $k$. The vector $vec{k}$ is defined as being rotated by Euler angles $alpha,beta$ from a fiducial momentum $vec{k}_{0}$ along the z-axis. The actual 3-momentum is,
$vec{k}=R(alpha,beta,0)vec{k}_{0}$ where $R(alpha,beta,gamma)$ is the rotation matrix and $alpha,beta,gamma$ are Euler angles. The Euler angle convention is $alpha$ is rotate about z, $beta$ is rotate about resultant y, $gamma$ is rotate about resultant z. The matrix $D^{j}(alpha,beta,0)^{m}_{ lambda}$ is Wigner's D-matrix. $lambda=pm 1$ is the photon's helicity. The states $|vec{k},lambdarangle$ are the linear-momentum-helicity eigenvectors. The states $|jmklambdarangle$ are angular-momentum-helicity eigenvectors. This result is equation (8.7.2), page 147 of thge book "Group Theory in Physics" by Wu-Ki Tung. It is also equation (28.35) on page 218 of the book "Relativistic Theory of Reactions", by J. Werle.

Vadim · Answer

I posted an answer to a similar question here. Admittedly, many different things may be implied when talking about the wave function of a photon. However, one should keep in mind is that, unlike electrons, photons are classically waves. Quantization neither adds nor subtracts from their wave-like properties, but injects discreteness (i.e. makes them from an electromagnetic field into countable photons). Their wave modes remain the same - plain waves of the electromagnetic field.

Answered by Vadim on January 10, 2021

Charles Francis · Answer

I define the photon wave function in a covariant formulation which has four polarisation states, two of which are not observable. Some authors use only transverse states, but the other two states would appear on Lorentz transformation, and they appear to be necessary to derive the classical correspondence correctly.

For momentum $p=(P^0,mathbf p)$, define a longitudinal unit 3-vector,
$$mathbf w(mathbf p,3) = frac {mathbf p} {|mathbf p |} $$
and orthogonal transverse unit 3-vectors $mathbf w(mathbf p,1),mathbf w(mathbf p,2)$ such that for $r,s= 1,2,3$
$$ mathbf w(mathbf p,r) cdot mathbf w(mathbf p,s) = delta_{rs} $$
Define normalised spin vectors,
$$ mathbf w(mathbf p,0) = (1,mathbf 0)$$
$$ mathbf w(mathbf p,r) = (0,mathbf w(mathbf p,r))$$
For momentum $p$ a photon plane wave state is given by the wave function,
$$langle x|p,rrangle = lambda(|mathbf p |,r) w(mathbf p,r) e^{-ix cdot p} $$
where $p^2 = 0$ and $ lambda $ is determined by relativistic considerations
$$ lambda(|mathbf p|, r) = frac 1{(2pi)^{3/2}} frac 1{sqrt{2p^0}} $$

You can then express a photon wave function as an integral

$$f^a(x) = frac 1{(2pi)^{3/2}}  = sumlimits_{r=0}^3 int frac{d^3mathbf p}{sqrt{2p^0}} mathbf w(mathbf p,r) e^{-ix cdot p} langle mathbf p, r|frangle$$

I took this from lecture notes at Cambridge, and I am not sure which books do things much the same way (there are some normalisation choices, as well as choice of gauge). I have given more detail in A Construction of Full QED Using Finite Dimensional Hilbert Space and in The Mathematics of Gravity and Quanta

Wavefunction of a photon

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