Is there a relation between quantum theory and Fourier analysis? I found that some theories about quantum theory is similar to Fourier transform theory. For instance, it says "A finite-time light's frequency can't be a certain value", which is similar to "A finite signal has infinite frequency spectrum" in Fourier analysis theory. I think that a continuous frequency spectrum cannot be measured accurately, which is similar to Uncertainty principle by Hermann Weyl. How do you think about this?
 A: Yes, there is a very strong interconnection.
A particle in q.m. doesn't have a defined position. Instead, there is a function describing the probability amplitude distribution for the position: the wavefunction $u(x)$. This is always told even in books for the general public. However, the momentum of the particle too isn't, in general, well defined: for it also we have a probability amplitude distribution, let's call it $w(p)$.
It happens that $u$ and $w$, the space- and momentum-wavefunctions, are in some sense Fourier transforms of each other. The reason is the following.
In Dirac's notation,
$$u(x) = \langle x|\psi\rangle,\quad w(p)=\langle p|\psi\rangle $$
where $|\psi\rangle$ is the state of the particle, $|x\rangle,|p\rangle$ are respectively the eigenstates of the position and momentum operators.
Let's say we work in the $x$ basis. The $p$ operator is written $-i\hbar\partial/\partial x$. To find eigenstates of $p$, we can call $\langle x|p\rangle=f_p(x)$ and this has to obey the eigenvalue equation for $p$ in the $x$ basis representation:
$$ -i\hbar\frac{\partial}{\partial x}f_p(x)=pf_p(x)$$
whose solution is the family of functions $f_p(x) = e^{ipx/\hbar}$.
Now, to change from a basis to the other we can write
$$\langle p|\psi\rangle= \int \langle p|x\rangle\langle x|\psi\rangle dx$$
or
$$ w(p) = \int e^{-ipx/\hbar}u(x)dx$$
which is a Fourier transform!
The $\hbar$ factor is to give the correct dimensionality.
Nice, isn't it?
As you pointed out, the fact that if $u$ is "spread" then $w$ is "peaked" and vice versa is typical of Fourier transformed functions.
So Heisenberg's principle can be thought to come from here.
This holds for a lot of conjugated quantum variables.
