How do we normalize a Dirac delta function position space wave function? I have a position space wavefunction $$\psi(x) = \delta(x-a) + \delta(x+a).$$ Now the question states to compute the following: 


*

*The Fourier transform of $\psi(x)$. (Which invariably is the momentum space wave function $\phi(p)$)

*Compute $\Delta p \Delta x$


Now to compute $\phi(p)$ one first need to normalize $\psi(x)$, but while doing so, one need to compute, $$|N|^2\int\limits_{-\infty}^{+\infty}  (\delta(x-a) + \delta(x+a))^2 dx =1,$$ $N$ being the normalization constant. But my question is how to go about computing this integral?
The normalization integral is now done, any help, in solving $\Delta x$, or $\Delta p$? I can't see through this problem?
 A: The correct normalization factor is $$ N = \frac{1}{\sqrt{2}}.$$
To see this, note that you can write your wave-function in ket notation as
$$\psi(x) = \langle x | a \rangle + \langle x | -a \rangle \equiv \langle x | \psi \rangle, $$
where we have used the usual basis for the (one dimensional) position representation, with normalization
$$ \langle x | y \rangle = \delta(x-y),$$
and we have defined the state $|\psi\rangle$ as the state corresponding to your wave function:
$$| \psi \rangle = | a \rangle + |-a \rangle.$$
Your question now translates into: what is $\langle \psi | \psi \rangle$?
The answer is readily obtained: this is a sum of two orthogonal states (assuming $a\neq0$), hence
$$ \langle \psi | \psi \rangle = \langle a | a \rangle + \langle -a | -a \rangle = 2$$
and
$$ | \psi \rangle = \frac{1}{\sqrt{2}} ( |a\rangle + |-a\rangle),$$
$$ \psi(x) = \frac{1}{\sqrt{2}} ( \delta(x-a)+ \delta(x+a)).$$
Note that if you try to compute $\langle a | a \rangle$ as an integral you get $\infty$:
$$ \langle a | a \rangle = \int dx \delta(x-a)^2 \approx \delta(0) \approx \infty $$
where the $\approx$ is because we are being mathematically sloppy here, see this discussion of Qmechanic about it.
This is due to the use of a continuous base. You can always think of it as the limiting case of a discrete base, taking case this way of the mathematical difficulties.
