Some of you may have seen my post yesterday which asked essentially the same question as I'm going to ask here and has since been deleted, for which I apologize. I noticed some egregious errors in my math and the way I phrased the question that made it seem better to try to start fresh.
I'm interested in trying to understand the physics of "single photons" in terms of quantum optics. Namely, I'm trying to understand how to express the state of a photon as a wave function in position space and to what physical scenarios such wave functions correspond. Here's what I've been looking at so far:
Reading through this paper, which has an EM field state given which the "wave function of a single photon" is part of. Equation (98), neglecting helicity and working only in 1-dimension, reads:
\begin{equation} \left|\Psi_{1}\right\rangle=\int d k f(k) a_{k}^{\dagger}|0\rangle\tag{1} \end{equation}
Where $\Psi_{1}$ is the "state consisting of a single photon" which I take to mean something like the Fock space state vector, $f(k)$ is the "wave function of a single photon" in what I would take to be a "photon Hilbert space" (forgive me if the terminology here is a little bit off), and $a_{k}^{\dagger}$ is the creation operator that creates a photon of momentum $k$.
This state corresponds to an EM field state containing all possible momenta. I was interested in the possibility of making this state "band-limited" such that the values of momenta are discrete and finite. I think that this would correspond to a physical situation like a "photon in a finite potential well". My suspicion is that such a state would be created by taking equation (1) and making it such that:
\begin{equation} f(k)=\sum_{n=0}^{N} c_{k} \delta(k-n)\tag{2} \end{equation}
Where of course $\delta(k-n)$ is the Dirac delta function and $n$ is an integer value with dimensions of $k$. So then plugging this in:
\begin{equation} \left|\Psi_{1}\right\rangle=\int d k \sum_{n=0}^{N} c_{n} \delta(k-n) a_{k_{n}}^{\dagger}|0\rangle=\sum_{n=0}^{N} c_{n}a_{k_{n}}^{\dagger}|0\rangle=\sum_{n=0}^{N} \frac{c_{n}}{\sqrt{n!}}|n\rangle\tag{3} \end{equation}
If, for whatever reason, we wanted to find the position space representation of this Fock state, I believe it would be:
\begin{equation} \Psi_{1}(x)=\langle{x}|\Psi_{1}\rangle=\sum_{n=0}^{N} \frac{c_{n}}{\sqrt{n!}}\langle{x}|{n}\rangle=\sum_{n=0}^{N} \frac{c_{n}}{\sqrt{n!}}\psi_n(x)\tag{4} \end{equation}
Where $\psi_n(x)$ is the $n^{th}$ position-space wave function of the quantum harmonic oscillator. The first of many things that isn't clear to me is whether this "position space representation of the Fock state $|\Psi_{1}\rangle$" has any physical meaning in and of itself. However, what I feel pretty confident does have a physical meaning in this scenario is the position space wave function of the photon itself $f(x)$ which is of course related to $f(k)$ by a Fourier transform:
\begin{equation} f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}f(k)e^{ikx}dk=\frac{1}{2\pi}\int_{-\infty}^{\infty}\sum_{n=0}^{N} c_{n} \delta(k-n)e^{ikx}dk=\frac{1}{2\pi}\sum_{k_{n}}^{k_{N}} c_{{k}_{n}} e^{i (k_{n} x-\omega_{k_{n}} t)}\tag{5} \end{equation}
Where $\omega_{k_{n}}$=$ck_{n}$ via the dispersion relation. So in other words, the photon wave function in this finite potential well is simply a sum over plane waves of definite momenta and frequencies, basically exactly as you'd expect.
So finally, my question is: does this derivation of a "photon in a box" wave function have any physical or experimental meaning? I've been trying to answer basically this exact question for a while. I've found a state very similar to my equation (5) in this paper in which their equation (2) reads:
\begin{equation} |\psi\rangle=|g\rangle \otimes \sum_{n=0}^{N} c_{n}|n\rangle\tag{6} \end{equation}
In the case of that paper, they are interested in synthesizing arbitrary superpositions of Fock states using a superconducting microwave resonator. I'd really like to understand if equation (6) has any relationship to equation (5) and if so, what that relationship is and how to understand it better. Is even the concept of a "photon in a box" with a wave function given by my equation (5) even experimentally viable in the real world?
