$V(r)=\frac{1}{r}$ means for any two electrons at position $r_1$ and $r_2$, the electric potential is given by $\frac{1}{|r_1-r_2|}$
The Fourier transform of $\frac{1}{r}$ is $\frac{1}{q^2}$.
How can we interpret $V(q)=\frac{1}{q^2}$ physically?
This is what I'm thinking about:
(1) $V(q)=\frac{1}{|q_1-q_2|^2}$ does not describe the potential between two electrons with wave vector $q_1$ and $q_2$
If we have two electrons with wave vector $q_1$ and $q_2$ (in othe words wavefunctions are $e^{iq_1r}$ and $e^{iq_2r}$), the probability distributions are uniform in space, then the potential $V(q)$ should be constant for all combination of $q_1$ and $q_2$.
(2) $V(q)$ must be quantum mechanics in nature ?
It is hard to explain in classical physics, but if we think about quantum Hilbert basis,
$$
\langle r_1',r_2'|\hat{V}|r_1,r_2\rangle =\frac{1}{|r_1-r_2|} \delta_{r_1,r_1'} \delta_{r_2,r_2'}$$
$$
\langle q_1',q_2'|\hat{V}|q_1,q_2\rangle =\frac{1}{|q_1-q_1'|^2} \delta_{q_1+q_2,q_1'+q_2'}
$$
$\hat{V}$ is two particles' operator. $\frac{1}{r}$ or $\frac{1}{q^2}$ are just the representation in certain basis.
Therefore $V(q)$ is an off-diagonal term, so it must be something quantum.
(3) what does the energy value $V(q)$ corresponds to?
$V(q)$ has the dimensional of energy, my naive understanding is, it corresponds to the energy you need to "push" $|q_1,q_2 \rangle$ to a new state $|q_1-q,q_2+q\rangle$.
In classical picture, both $|q_1,q_2 \rangle$ and $|q_1-q,q_2+q\rangle$ configuration, the electrons are uniformly distributed, no work need to be done to change the Coulomb potential.
In quantum mechanics, let's take the analogy of two level system $$|\uparrow\rangle :=|q_1,q_2 \rangle $$
$$|\downarrow\rangle := |q_1-q,q_2+q\rangle $$
Then the off-diagonial term $\Delta=\langle \uparrow |\hat{V}|\downarrow \rangle =\frac{1}{q^2} $ is more like tunnelling rate ( hopping rate ~ frequency ~ energy).
So, "push" means tunnelling. The energy value $V(q)$ is tunnelling rate.
People are talking about $\frac{1}{q^2}$ divergence, I don't understand it. Does it mean two electrons exchange their momentum very often, and the rate is divergent for small momentum ($q\rightarrow 0$) exchanges?
(4) $V(q)$ as a Laplace transformation in classical E&M
Contradict to (2), we do have $V(q)$ in purely classical systems.
This often solve the problem of screened plasma potential, classical Drude model of metal, etc..
$$
\nabla^2 \phi(r)=\delta(r)
$$
The Laplace transformation gives :
$$
q^2 \phi(q)=1
$$
And $q$ does not mean the momentum of electron wavefunction $e^{iqr}$ (there is no such concept of wavefunction in classical physics
). $q$ means the oscillation of charge density:
given charge density wave $\rho(r)=\cos(qr)$, we have potential wave $V(r) = \frac{1}{q^2}\cos(qr)$
Classical: a point charge can be represented by a superposition of cosine charge waves from all wave-vector.
$$
\rho(x)=\delta(x-r)=\sum_q \cos(q(x-r))
$$
Quantum: a point charge state can be represented by a superposition of momentum q eigen-waves from all wave-vector.
$$
|r\rangle =\sum_q e^{iqr}|q\rangle
$$
I couldn't find a natural way to connect quantum and classical. Firstly, classical density $\cos(qr)$ changes it signs, while quantum $e^{iqr}$ is uniform in density; secondly, $\sum_q$ is coherent superpositions in the quantum case. This might be a bad analogy.
What is your interpretation of $V(q)=\frac{1}{q^2}$?
Best Answer
Sometimes it happens that in a pursuit of ultimate understanding we overcomplicate simple concepts.
What you should focus on is that:
A Fourier transform is a mathematical procedure that allows to expand a given function (continuous, differentiable etc.) into a complete set of orthonormal functions, usually utilizing a plane-wave kernel: $$F'(x) = \int dq \; F(q) e^{i q \cdot x}$$ It does not need to be more complicated than this.
Why would one use FT-functions instead of original ones? In many cases it makes math simpler. A good example is Maxwell's equations. In the position basis, they have a complicated form of first-order vector differential equations, while in momentum space they are just linear equations. $$\nabla \cdot \pmb E(\pmb r,t) = \rho(\pmb r,t) \;\; \rightarrow \;\; i \pmb k \cdot \pmb E' = \rho$$ where prime denotes a Fourier-transformed function. You can work out the rest yourself, if curious.
Sometimes scientists assign physical meaning to the FT'd functions, but it happens rarely, up to my knowledge (signal theory, information processing etc.). Most of the time, speaking of FT'd values, they actually mean the observables in any space one is comfortable with, because one can always back-transform (if nothing is seriously messed up).
My humble opinion in this case: you are thinking about that too much. An FT of the potential is just an FT of the potential. Since the mapping between V(r) and FT'd V'(q) is one-to-one and onto (in your case and in many other physical ones), you can think of V'(q) as a mathematical and physical equivalent of V(r).