Fourier Representation of Dirac's Delta Function

Question

This question is related to this other question on Phys.SE.

In quantum mechanics is often useful to use the following statement:

$$\int_{-\infty}^\infty dx\, e^{ikx} = 2\pi \delta(k)$$

where $\delta(k)$ is intended to represent Dirac's Delta Function. I would like to understand this statement, or at least know a justification of it, rather than blindly apply this result. From what I currently understand about this topic the equation above should be the Fourier representation of the Dirac's Delta Function, however I don't see how to prove it. Furthermore, since the Delta Function is not even a function, this statement appears to me as really strange.

Keep in mind that I am no expert on this topic, and an elementary explanation is what I seek. I would prefer a proof suited for an undergraduate student rather than a really rigorous and complex one.

Answer 1

I'll put a rigorous explanation first, then a loosey-goosey one afterwards.

This is all rooted in distribution theory. I'll work in $\mathbb{R}^n$ and use the convention that the Fourier transform has a $(2\pi)^{-n/2}$ out front (making it unitary), as well as the more standard sign. That is, $$\mathcal{F}f(\xi)=(2\pi)^{-n/2}\int\limits_{\mathbb{R}^n}f(x)e^{-ix\cdot\xi}\, dx$$

The Dirac delta is an example of a tempered distribution, a continuous linear functional on the Schwartz space. We can define the Fourier transform by duality: $$\langle\mathcal{F} u,\varphi\rangle=\langle u,\mathcal{F}\varphi\rangle$$ for $u\in\mathcal{S}'$ and $\varphi\in\mathcal{S}.$ Here, $\langle \cdot,\cdot\rangle$ denotes the distributional pairing. In particular, the Fourier inversion formula still holds. So, for $u=\delta,$ $$\langle\mathcal{F}\delta, \varphi\rangle=\langle\delta,\mathcal{F}\varphi\rangle=\mathcal{F}\varphi(0)=\langle (2\pi)^{-n/2},\varphi\rangle\implies \mathcal{F}\delta=(2\pi)^{-n/2}.$$ Now, the inversion formula gives that $$(2\pi)^{n/2}\delta=\mathcal{F}1,$$ and $\mathcal{F}1$ "equals" $$(2\pi)^{-n/2}\int\limits_{\mathbb{R}^n}e^{-ix\cdot \xi}\, dx$$ (sign in the exponential doesn't matter here). This is what you wrote if $n=1$.

Since you also wanted a less rigorous answer, this is how you might see it done in physics books:

Loosely, $$\mathcal{F}\delta(\xi)=(2\pi)^{-1/2}\int\limits_{-\infty}^\infty \delta(x)e^{-ix\xi}\, dx=(2\pi)^{-1/2}e^{-ix\xi}|_{x=0}=(2\pi)^{-1/2},$$ so "Fourier inversion" gives

$$\delta(x)=(2\pi)^{-1/2}\int\limits_{-\infty}^\infty \mathcal{F}\delta(\xi)e^{ix\xi}\, d\xi=(2\pi)^{-1}\int\limits_{-\infty}^\infty e^{ix\xi}\, d\xi.$$

Of course, these formal calculations are made rigorous by doing what I initially wrote.

Answer 2

Let the Fourier transform of a function $f$ be $$ \mathcal{F}\{f(x)\} = \int_{-\infty}^{\infty} f(x) \, e^{-ikx} \, dx. $$

Then the Fourier transform of the Dirac delta-function (well, actually it's not a function, but the calculations work anyways) is $$ \mathcal{F}\{\delta(x)\} = \int_{-\infty}^{\infty} \delta(x) \, e^{-ikx} \, dx = 1. $$

According to the Fourier inversion theorem, if $\mathcal{F}\{f(x)\} = F(k)$ then $\mathcal{F}\{F(x)\} = 2\pi\,f(-k).$ Applying this, we get $$ \int_{-\infty}^{\infty} e^{-ikx} \, dx = \int_{-\infty}^{\infty} 1(x) \, e^{-ikx} \, dx = \mathcal{F}\{1(x)\} = 2\pi\,\delta(k) . $$ By symmetry we also have $$ \int_{-\infty}^{\infty} e^{ikx} \, dx = 2\pi\,\delta(k) . $$

Answer 3

I thought that it might be instructive to present a way forward that uses a regularization of the Dirac Delta. To that end we proceed.

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PRELIMINARIES:

Let $\displaystyle \delta_L(k)=\frac1{2\pi}\int_{-L}^Le^{ikx}\,dx$. Then, we can write

$$\delta_L(k)=\begin{cases}\frac{\sin(kL)}{\pi k}&,k\ne0\\\\\frac L\pi&,k=0\tag1\end{cases}$$

The function $\delta_L(k)$ has the following properties:

1. For each $L$, $\delta_L(k)$ is an analytic function of $k$.
2. $\lim_{L\to \infty} \delta_L(0)= \infty$
3. $\left|\int_{-\infty}^x \delta_L(k')\,dk'\right|$ is uniformly bounded.
4. $\lim_{L\to \infty}\int_{-\infty}^{k}\delta_L(k')\,dk'=u(k)$, where $u$ is the unit step funciton.
5. For each $L>0$, $\int_{-\infty}^\infty \delta_L(k)\,dk=1$

While heuristically $\delta_L(k)$ "approximates" a Dirac Delta when $L$ is "large," the limit of $\delta_L(k)$ fails to exist. However, if we interpret this limit in the distributional sense, then $\lim_{L\to\infty}\delta_L(k)\sim\delta(k)$. We will now show that this is indeed that case.

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ANALYSIS:

Let $\phi(k)\in S$ where $S$ is the Schwarz Space of functions.

We will now evaluate the limit

$$\begin{align} \lim_{L\to \infty}\int_{-\infty}^\infty \delta_L(k)\phi(k)\,dk=\lim_{L\to \infty}\int_{-\infty}^\infty \frac{\sin(kL)}{\pi k}\phi(k)\,dk\tag1 \end{align}$$

Integrating by parts the integral on the right-hand side of $(1)$ with $u=\phi(k)$ and $v=\int_{-\infty}^{kL}\frac{\sin(x)}{\pi x}\,dx$ reveals

$$\begin{align} \lim_{L\to \infty}\int_{-\infty}^\infty \delta_L(k)\phi(k)\,dk&=-\lim_{L\to \infty}\int_{-\infty}^\infty \phi'(k)\int_{-\infty}^{kL}\frac{\sin(x)}{\pi x}\,dx\,dk\tag2 \end{align}$$

Using Property 3 in the Preliminaries section, there exists a number $C$ such that $\left|\phi'(k)\int_{-\infty}^{kL}\frac{\sin(x)}{x}\,dx\right|\le C\,|\phi'(k)|$. Inasmuch as $C|\phi'(k)|$ is integrable, the Dominated Convergence Theorem guarantees that

$$\begin{align} \lim_{L\to \infty}\int_{-\infty}^\infty \delta_L(k)\phi(k)\,dk&=-\lim_{L\to \infty}\int_{-\infty}^\infty \phi'(k)\int_{-\infty}^{kL}\frac{\sin(x)}{\pi x}\,dx\,dk\tag3\\\\ &=-\int_{-\infty}^\infty \phi'(k)\lim_{L\to \infty}\left(\int_{-\infty}^{kL}\frac{\sin(x)}{\pi x}\,dx\right)\,dk\\\\ &=- \int_{-\infty}^\infty \phi'(k)\underbrace{u(k)}_{\text{Unit Step}}\,dx\\\\ &=-\int_0^\infty \phi'(k)\,dk\\\\ &=\phi(0) \end{align}$$

Therefore, in the sense of distributions as given by $(3)$, we assert that $\lim_{L\to\infty}\delta_L(k)\sim \delta(k)$ whereby rescaling yields the distributional limit

$$\bbox[5px,border:2px solid #C0A000]{\lim_{L\to \infty}\int_{-L}^Le^{ikx}\,dx\sim 2\pi \delta(k)}$$

as was to be shown!