Is it correct to describe the emission of a single photon from an atom as a spherically-expanding wave function, the absolute square of which at any instant in time furnishes the probability of finding a photon at some point on the surface at that instant? If so, what is the relationship between the description of the propagating wave packet associated with the photon itself and the spherically-expanding wave function?
Answer
Let us first assume we are the simpler context of quantum mechanics (rather than field theory). Let us assume as well that photons obey Schrödinger's equation (which is not exactly correct) for the sake of simplicity. So we attempt to solve the equation
\begin{equation}\hat{H}|\psi\rangle = i\hslash \frac{\partial}{\partial t}|\psi\rangle\tag{1}\label{eq:Schrod}\end{equation}
to obtain its wave function, $|\psi\rangle$. So a wave function is a normalized solution for Schrödinger's equation (in this setting).
There are however many techniques to solve such a system depending on the case at hand. One of the methods is to employ a Fourier transformation and solve the equation in terms of momentum. Additionally, one could also look for stationary solutions, solutions whose time dependence is written as $e^{-\frac{i}{\hslash}Et}$. These might be the most common or basic treatments and they lead to the idea of studying the equation on a "per mode" approach. That is, one focuses on finding a solution for a fixed momentum $k$ or energy $E$. However, it is usually forgotten that these are partial solutions to more realistic scenarios, where the photon (actually photon field) is likely composed by many solutions of this sort with many different momenta. It turns out that the linearity of the equation \eqref{eq:Schrod}, tells us that those superpositions of solutions, namely sums (or integrals) are also solutions to it. This superpositions are what is known as wave packages. Mathematically they should look something like $$\Psi(x) \sim \sum_k c_k\psi_k(x)$$ which resembles the Fourier transform. The picture one has in mind is that of a localized bump whose shape is given by the particular coefficients $c_k$ packaging the individual modes (the fixed $k$ wave functions).
I hope that clears up the difference. About the spherical wave, it is just a particular set of coordinates for the problem, but it doesn't change anything of what I have described. To finish I would like to point you perhaps towards the topic of quantum field theory in particular to QED (quantum electrodynamics) which is the theory that properly explains the propagation and interaction of photons with matter.
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