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?
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