# Plane wave basis in a box. Inconsistent terminology about Fourier series and transform

I will refer to the one-dimensional equivalent of the problem in the picture for simplicity. At the end of the picture, the author argues that the Fourier transform of the function $$\psi(x)$$ is the expression for the coefficients $$a_k$$ of the Fourier series. Is this a valid and general statement?

The notes seem to be mixing the concepts of Fourier series and Fourier transform. $$\psi$$ is argued to be writable as a Fourier series, does it mean that $$\psi$$ is periodic? If it were, the coefficients expresion of the Fourier expansion of a periodic function $$f$$ with period $$L$$ should follow a different formula: $$f(y) =\sum_{k} c_k e^{iky}$$ $$c_k =\frac{1}{L}\int_{-L/2}^{L/2} dy f(y) e^{-iky}\tag{1}$$

from that of the Fourier transform of a non-periodic function $$g$$:

$$\hat{g}=h_1\int_{-\infty}^{\infty}dy g(y) e^{-iky} \tag{2}$$ (where $$h_1$$ is some constant according to the adopted convention, such that the respective constant $$h_2$$ in the expresion for the inverse formula should verify $$h_1h_2=\frac{1}{2\pi}$$) which clearly doesn't match with (1)

So translating the author's statement to 1 dimension and pluging the expresion of $$\phi_k(x)$$:In the problem V is a cubic box $$[-L/2,L/2]^3$$, so in 1 dimension it is the segment $$[-L/2,L/2]$$,
$$\psi(x)=\sum_{k} a_k \phi_k(x) =\sum_{k} a_k e^{ikx}$$ and $$a_k=(\phi_k,\psi) =\int_{[-L/2,L/2]} dy \phi_k^*(y)\psi(y)= \frac{1}{\sqrt{L}}\int_{-L/2}^{L/2}dy \psi(y) e^{-iky} \tag{1'}$$

From the stament of the author, I am confused about

a) If $$a_k$$ are the coefficients of a Fourier series, why don't we have a $$\frac{1}{L}$$ factor in (1') , just like in (1) instead of $$\frac{1}{\sqrt{L}}$$ ? Besides I don think $$psi$$ is periodic, even though the formula looks pretty much alike

b) If at the same time $$a_k$$ is the Fourier transform of $$\psi$$,from the definition of Fourier transform , I should have something like $$a_k=\hat{\psi}=h_1\int_{-\infty}^{\infty}dy \psi(y) e^{-iky}\tag{1''}$$, which also fails to agree with (1) or (1') because of the limits of integration and because of the constant h_1 which is normally taken to be $$h_1=1$$( implying $$h_2=1/2\pi$$) an not $$\frac{1}{\sqrt{L}}$$

Can someone clear this up? However, one often uses the trick of normalization in a box, assuming that the system of interest is continued periodically beyond the interval of interest (the box $$[-L/2,L/2]$$), expanding it in Fourier series, and then taking the size of the system (i.e., the period of the Fourier series) to infinity. The sum over momentum vectors $$k$$ then becomes an integral and the measure used in this integral integral is known as the density-of-states (DOS).
• In the problem of plane waves over all of $\mathbb{R}$ however, the Fourier transform of the function that yields the coefficient does match the mathematical one since the limits are now from $-\infty$ to $\infty$. Jan 12, 2021 at 15:10