Converting an expression of a particular k-mode to the spatial domain

AI Thread Summary
The discussion centers on converting the expression for k-mode density, n_k, into the spatial domain using inverse Fourier transforms. The initial approach involves integrating terms related to n(x) and a(x) but leads to uncertainty in the next steps. An alternative method suggests multiplying both sides of the equation by exponential terms, but this also raises doubts about correctness. The participant notes that the inverse Fourier transform of a product results in a convolution, indicating a potential misunderstanding in their approach. Overall, the conversation highlights challenges in applying Fourier transform techniques to achieve the desired spatial representation.
ian_dsouza
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Homework Statement
This is a problem in cosmology but I'll boil it down to the math. $n(\vec{x})$ is the (spatial) number density of particles. By the Fourier transform, $$n(\vec{x}) = \int\mathrm{d}^3k\ n_\vec{k}e^{i\vec{k}\cdot\vec{x}}$$. Also, $a(\vec{x})$ is the field representing the particle (a concept stemming from quantum field theory). By the Fourier transform, $$a(\vec{x}) = \int\mathrm{d}^3k\ a(\vec{k}) e^{i\vec{k}\cdot\vec{x}}$$.



It is known that $$n_\vec{k} = \omega a^2(\vec{k})$$, where $$\omega = \frac{k}{R}$$ and $R$ is a function of time only. I want to find an expression for $n(\vec{x})$ in terms of $a(\vec{x})$ and I was wondering if someone could help me out.
Relevant Equations
$$n(\vec{x}) = \int\mathrm{d}^3k\ n_\vec{k}e^{i\vec{k}\cdot\vec{x}}$$
$$a(\vec{x}) = \int\mathrm{d}^3k\ a(\vec{k}) e^{i\vec{k}\cdot\vec{x}}$$
$$n_\vec{k} = \omega a^2(\vec{k})$$
$$\omega = \frac{k}{R}$$
$$n_\vec{k} = \omega a^2(\vec{k})\tag{1}$$
One way is to write the inverse Fourier transforms of the terms above. So, eqn (1) becomes
$$\int\mathrm{d}^3x\ n(\vec{x})e^{-i\vec{k}\cdot\vec{x}} = \omega \int\mathrm{d}^3x^\prime\ a(\vec{x^\prime})e^{-i\vec{k}\cdot\vec{x^\prime}} \int\mathrm{d}^3x^{\prime\prime}\ a(\vec{x^{\prime\prime}})e^{-i\vec{k}\cdot\vec{x^{\prime\prime}}}$$
But I don't know how to proceed from here.

On the other hand, I could multiply both sides of eqn (1) by $$e^{i\vec{k}\cdot\vec{x}}e^{i\vec{k^\prime}\cdot\vec{x}}$$ and first writing $$a^2(\vec{k}) = a(\vec{k})a(\vec{k^\prime})$$ (don't know if this is right though).
$$n_\vec{k}e^{i\vec{k}\cdot\vec{x}}e^{i\vec{k^\prime}\cdot\vec{x}} = \omega a(\vec{k})a(\vec{k^\prime})e^{i\vec{k}\cdot\vec{x}}e^{i\vec{k^\prime}\cdot\vec{x}}$$
Integrating with respect to $\vec{k}$ and $\vec{k^\prime}$
$$\int\mathrm{d}^3k\ n_\vec{k}e^{i\vec{k}\cdot\vec{x}}\int\mathrm{d}^3k^\prime\ e^{i\vec{k^\prime}\cdot\vec{x}} = \omega \int\mathrm{d}^3k\ a(\vec{k})e^{i\vec{k}\cdot\vec{x}}\int\mathrm{d}^3k^\prime\ a(\vec{k^\prime})e^{i\vec{k^\prime}\cdot\vec{x}}$$
$$n(\vec{x})\delta^3({\vec{x}}) = \omega a^2(\vec{x})$$
But this isn't the answer I am looking for either and I suspect I am doing something wrong.
 
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