Periodicity of function as a result of superposition in Quantum Mechanics Say we add infinitely many waves (states of definite momentum) so as to produce a function that gives a very well-defined position, does that addition(using Fourier series) make that function periodic? I mean, if the added waves are themselves periodic(in Fourier they are harmonic), does that mean that the resulting function will also be periodic?
If so, in the case of the superposition of states that give us a Dirac Delta function, does that mean that the delta function will reappear after some long distance or is it just a single spike in all space?
 A: I'd like to pick apart a few things. First, I'll echo CuriousOnes comment that a state of well-defined position, represented in position space as delta function $\delta (x-x_0)$ and in momentum space as a plane-wave, is not a true physically realizable state. It is at least a nice mathematical device, however, and we can roughly think of it as an approximation for a very narrow wave-packet.
That important caveat out of the way, let's work in the position basis spanned by the states $|{x}\rangle$, where
$$\langle x'| x\rangle = \delta(x-x').$$
This delta function can be written as the Fourier transform of plane-wave via
$$\psi(x) = \delta(x-x_0) = \int \frac{dk}{{2\pi}}e^{-i k x}e^{i k x_0},$$
so we can think of this "state" as being a linear superposition of all the plane-wave states $e^{-ikx}$ with equal probabilities (but varying phases given by $e^{i k x_0}$. From a Fourier analysis perspective, we can think of this as a superposition of oscillatory functions of wavelength $\lambda = 2\pi/k$. Since $k$ can be any real number, $k$ can be any real number, and so in this expansion there exist wavelengths that are irrational multiples of each other, making it so that this function is not periodic.
A quick aside
If you add two sine functions
$$f(x) = \sin(k_1 x) + \sin(k_2 x)$$,
then the resulting function is periodic if and only if the $k$'s form a rational ratio. To see this, note that in order for this function to be periodic, both sine functions must go through an integer multiple of cycles in the same $\Delta x$.
Extending this idea, in order for an arbitrary sum of oscillatory functions to be a periodic function, there must be some minimum $k$ that all the individual $k$'s are multiples of.
Thus
If you form an equal superposition of plane waves in the form of a Fourier series, rather than a Fourier transform as above, you get a periodic function. That is,
$$\sum_{n=-\infty}^{\infty} e^{i2\pi x/\lambda_0} = \sum_{m=-\infty}^{\infty}\delta(x-m\lambda_0),$$
where $\lambda_0$ is the minimum wavelength of the plane-waves that show up and hence is the wavelength of the resulting function. The resulting "function" is known as a Dirac comb and consists of an infinite array of equally spaced delta functions.
The Dirac comb is actually used in physics: it shows up in simple models for solids displaying a band structure, and it shows up in the description of the frequency comb, for which John Hall of CU Boulder and NIST won a Nobel prize.
