The Delta Functional (`Delta Function')

The parameter $\sigma$ in the Gauss function determines its width. A broad wave packet in coordinate ($x$) space corresponds to a narrow distribution of wave vectors $k$. What happens in the limit $\sigma\to 0$ ? In coordinate space, this would correspond to an extremely sharp wave packet around $x=0$ that could serve as a model for a particle localized at $x=0$. We define

$$\fbox{$ \displaystyle \delta(x) \equiv\lim_{\sigma\to 0} \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}}.$}$$ (48)

As an ordinary function, this is a somewhat strange mathematical object because it is zero for all $x\ne 0$, but infinite for $x=0$. However, it has the useful property that for any (reasonably well-behaving) function $f(x)$

$$\int_{-\infty}^{\infty}dx' \delta(x-x')f(x')=\lim_{\sigma\to 0} \... ...}dx' f(x') \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-x')^2}{2\sigma^2}} = f(x).$$ (49)

For example, multiplication of $f(x')$ with $\delta(x')$ and integration over the whole $x'$-axis gives the value of $f$ at $x=0$. Such an operation is called a functional, that is a mapping

$$\delta: f \to f(0)$$ (50)

that puts a whole function $f$ to a (complex or real) number. Nevertheless, for historical reasons physicists call this object a delta-function. Remember that $\delta(x)$ is only defined as in (1.49), that is by integration over a function (`test-function') $f(x)$.

Another very useful property is the Fourier transform of the Delta-function: We recall our definition

$$f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}dk \tilde{f}(k) e^{ikx},\quad \tilde{f}(k):=\int_{-\infty}^{\infty}dx'{f}(x') e^{-ikx'}$$

$$\leadsto f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}dk \int_{-\infty}^{\infty}dx... ...infty}dx' \underline{\int_{-\infty}^{\infty}\frac{dk}{2\pi} e^{ik(x-x')}}f(x').$$ (51)

Now, comparing with the definition of the Delta function, Eq.(1.49), we recognise

$$\fbox{$ \displaystyle \delta(x-x') = \int_{-\infty}^{\infty}\frac{dk}{2\pi} e^{ik(x-x')}.$}$$ (52)

The delta function is thus a superposition of all plane waves; the corresponding distribution of $k$-values in $k$-space is `extremely broad', that is uniform from $k=-\infty$ to $k=+\infty$. Note that we can also obtain the result $\tilde{\delta}(k)=1$ from the Fourier transform $\tilde{g}(k)$ of the Gauss function $g(x)$, Eq. (1.46), in the limit $\sigma\to 0$.