Dirac Delta Function

⚡ Key Takeaways

The Dirac delta function $\delta(x)$ is a "generalized function" that is zero everywhere except at $x=0$ , where it is infinitely spiked so that its total integral equals $1$ . Its defining property is the sifting property: $\int_{-\infty}^{\infty} f(x)\delta(x-a)\,dx = f(a)$ . It is used throughout physics and engineering to model point masses, point charges, impulses, and idealized sources, and as the identity element of convolution. Introduced by Paul Dirac (~1927) in quantum mechanics, rigorously founded by Laurent Schwartz (1945) via distribution theory.

💡 The Core Insight

The delta function is best understood not by its pointwise values but by what it does: it is the evaluation functional. Feeding $\delta$ a test function $f$ returns $f(0)$ . Everything else — the spike, the unit area, the limit of tall thin bumps — is a picture meant to help intuition. The mathematically honest statement is: $\delta$ is the limit (in a weak sense) of unit-area "nascent delta" sequences, and its meaning lives entirely under an integral sign.

DefinitionDirac Delta — Heuristic Form

The Dirac delta function $\delta(x)$ is informally characterized by the two conditions

1

\quad \delta(x) = 0 \ \text{ for } x \neq 0, \qquad \delta(0) = +\infty,

Derivation hint: Not a derivation — this is the informal "infinite spike at the origin" picture; it is not literally valid as a Lebesgue function.

2

\quad \int_{-\infty}^{\infty} \delta(x)\,dx = 1.

Derivation hint: Normalization condition; the total area (mass) under the spike is defined to be unity. Here $x \in \mathbb{R}$ is the independent variable, and the integral is taken over the whole real line.

⚠️ Common mistake

Equations [1]–[2] are mutually inconsistent in classical analysis: a function that is zero except at a single point has Lebesgue integral $0$ , never $1$ . So $\delta$ is not a function. The honest definition is the functional one below.

DefinitionDirac Delta — Distributional (Rigorous) Form

Let $\mathcal{D}(\mathbb{R})$ be the space of test functions: smooth functions $\varphi:\mathbb{R}\to\mathbb{R}$ with compact support. The Dirac delta is the linear functional $\delta:\mathcal{D}(\mathbb{R})\to\mathbb{R}$ defined by

3

\quad \langle \delta, \varphi\rangle \;=\; \varphi(0).

Derivation hint: $\delta$ is defined as the evaluation-at-zero functional; $\langle\cdot,\cdot\rangle$ denotes the action of a distribution on a test function. More generally, the shifted delta $\delta_a = \delta(x-a)$ satisfies the sifting property

4

\quad \int_{-\infty}^{\infty} f(x)\,\delta(x-a)\,dx \;=\; f(a),

Derivation hint: Apply the evaluation functional at the point $x=a$ ; the integral notation is formal shorthand for $\langle\delta_a,f\rangle$ . valid for every continuous $f$ . Here $a\in\mathbb{R}$ is the location of the spike.

TheoremKey Properties of the Delta Function

For constants $a,b$ with $b\neq 0$ , and suitable test functions, the delta satisfies:

5

\quad \delta(x) = \delta(-x) \quad(\text{even}), \qquad x\,\delta(x) = 0.

Derivation hint: Substitute $x\to -x$ in the sifting integral; for $x\delta(x)$ , note $\langle x\delta,\varphi\rangle=\langle\delta,x\varphi\rangle=(x\varphi)|_{x=0}=0$ .

6

\quad \delta(b x) = \frac{1}{|b|}\,\delta(x).

Derivation hint: Change variables $u=bx$ in $\int f(x)\delta(bx)dx$ ; the Jacobian $|b|$ and the sign-handling of limits produce the $1/|b|$ factor.

7

\quad \delta\big(g(x)\big) = \sum_{i}\frac{\delta(x-x_i)}{|g'(x_i)|}, \quad g(x_i)=0,\ g'(x_i)\neq 0.

Derivation hint: Linearize $g$ near each simple root $x_i$ via $g(x)\approx g'(x_i)(x-x_i)$ and apply [6] to each isolated zero.

8

\quad \int_{-\infty}^{x}\delta(t)\,dt = H(x),

Derivation hint: Integrating the unit-area spike accumulates from $0$ to $1$ as $x$ crosses the origin, giving the Heaviside step function $H$ . where $H$ is the Heaviside step function, so symbolically $H'(x)=\delta(x)$ .

ProofDelta as a Limit of Nascent Deltas

We show that a sequence of ordinary, unit-area functions concentrating at the origin reproduces the sifting property in the limit.

Step 1. Choose a family $\delta_\epsilon(x)$ , e.g. the Gaussian

9

\quad \delta_\epsilon(x) = \frac{1}{\epsilon\sqrt{\pi}}\,e^{-x^2/\epsilon^2}.

Derivation hint: A normalized Gaussian of width $\sim\epsilon$ and height $\sim1/\epsilon$ ; its area is $1$ for every $\epsilon$ by the standard Gaussian integral.

Step 2. Verify unit area for all $\epsilon>0$ :

10

\quad \int_{-\infty}^{\infty}\delta_\epsilon(x)\,dx = \frac{1}{\epsilon\sqrt\pi}\int_{-\infty}^{\infty}e^{-x^2/\epsilon^2}dx = \frac{1}{\epsilon\sqrt\pi}\cdot \epsilon\sqrt\pi = 1.

Derivation hint: Substitute $u=x/\epsilon$ and use $\int e^{-u^2}du=\sqrt\pi$ .

Step 3. Take a continuous $f$ . By unit area,

11

\quad \int_{-\infty}^{\infty} f(x)\,\delta_\epsilon(x)\,dx - f(0) = \int_{-\infty}^{\infty}\big(f(x)-f(0)\big)\delta_\epsilon(x)\,dx.

Derivation hint: Subtract $f(0)=f(0)\int\delta_\epsilon\,dx$ and merge into one integral.

Step 4. As $\epsilon\to 0^+$ , $\delta_\epsilon$ concentrates near $x=0$ , where $f(x)-f(0)\to 0$ by continuity; the right side vanishes. Hence

12

\quad \lim_{\epsilon\to 0^+}\int_{-\infty}^{\infty} f(x)\,\delta_\epsilon(x)\,dx = f(0) = \langle\delta,f\rangle.

Derivation hint: Split the integral into $|x|<\eta$ (small by continuity) and $|x|\ge\eta$ (small by rapid decay of $\delta_\epsilon$ ); both $\to0$ . This is exactly the sifting property [4] with $a=0$ , justifying $\delta=\lim_{\epsilon\to0^+}\delta_\epsilon$ in the weak (distributional) sense. $\qquad\blacksquare$

∎

PA

Paul A. M. Dirac

1902–1984

Paul Dirac, one of the founders of quantum mechanics, introduced the delta function around 1927 in his work on transformation theory and quantum dynamics. He needed a continuous analogue of the discrete Kronecker delta $\delta_{ij}$ to express orthonormality of continuum eigenstates (e.g. position eigenstates) and to write resolutions of the identity. He treated $\delta(x)$ operationally — "an improper function" useful as a tool — fully aware it lacked classical rigor, writing that it could "be used as though it were a continuous function." His pragmatic confidence was vindicated two decades later when Laurent Schwartz built the theory of distributions, giving $\delta$ a precise mathematical foundation and earning Schwartz the 1950 Fields Medal.

🧠 Intuition

Picture squeezing a fixed amount of clay (mass $1$ ) into an ever-narrower, ever-taller column. The area never changes, but the shape limits toward an infinitely thin spike. The delta function is what survives in the integral: it remembers only "there is one unit of mass, located at $x=0$ ." When you multiply by a smooth function $f$ and integrate, the spike samples $f$ at exactly that location — like a probe so sharp it touches the curve at a single point and reports the height there. This is why $\delta$ is the identity for convolution: convolving any signal with $\delta$ leaves it unchanged, because $\delta$ is a "do-nothing" probe that just relays each value back.

Figure

A nascent-delta family

\delta_\epsilon(x)

collapsing to the spike at

x=0

. As

\epsilon

shrinks, the bumps grow taller and narrower while preserving unit area.

ExampleSifting with a Shifted Delta

Given. Evaluate $\displaystyle I = \int_{-\infty}^{\infty} \cos(\pi x)\,\delta(x-2)\,dx$ .

Step 1. The delta is centered at $a=2$ . By the sifting property [4],

\int_{-\infty}^{\infty} f(x)\,\delta(x-2)\,dx = f(2).

Step 2. Here $f(x)=\cos(\pi x)$ , so $f(2)=\cos(2\pi)=1$ .

Step 3. The point $x=2$ lies inside $(-\infty,\infty)$ , so the spike is captured.

\boxed{\,I = 1\,}\quad(\text{dimensionless})

ExampleScaling and Composition — Delta of a Function

Given. Evaluate $\displaystyle J = \int_{-\infty}^{\infty} e^{x}\,\delta(3x-6)\,dx$ .

Step 1. Write $g(x)=3x-6$ , whose only root is $x_0=2$ , with $g'(x)=3$ , so $|g'(x_0)|=3$ . By property [7],

\delta(3x-6) = \frac{\delta(x-2)}{|3|} = \tfrac13\,\delta(x-2).

Step 2. Substitute and pull out the constant:

J = \tfrac13\int_{-\infty}^{\infty} e^{x}\,\delta(x-2)\,dx.

Step 3. Apply sifting at $x=2$ : $\int e^{x}\delta(x-2)\,dx = e^{2}$ .

Step 4. Combine:

J = \tfrac13 e^{2} \approx \tfrac13(7.389) \approx 2.463.

\boxed{\,J = \tfrac13 e^{2} \approx 2.463\,}\quad(\text{dimensionless})

⚠️ Common mistake

" $\delta(0)$ is a number you can plug in." It is not. $\delta$ has no well-defined pointwise value; writing $\delta(0)=\infty$ is shorthand, never an actual quantity to compute with. Any expression that requires evaluating $\delta$ at a point without integrating is meaningless. Likewise, products of two deltas at the same point — e.g. $\delta(x)^2$ or $\delta(x)\delta(x)$ — are undefined, since you cannot multiply distributions in general.

Counterexample

A tempting but false manipulation: "Since $\delta(x)=0$ for $x\neq0$ and the value at one point doesn't affect an integral, $\int f(x)\delta(x)\,dx=0$ ." This reasoning fails because $\delta$ is not a Lebesgue-integrable function — it is a distribution. The "value $0$ everywhere except a point" heuristic is exactly the part that cannot be made into a classical integrand. The correct result is $\int f(x)\delta(x)\,dx = f(0)$ , which is generally nonzero. The lesson: never reason about $\delta$ as if it were an ordinary function.

connection

The delta function ties together many areas. Its antiderivative is the Heaviside Step Function, and its derivatives $\delta'(x)$ (the "doublet") form a tower of higher-order distributions used to model dipoles. The discrete analogue is the Kronecker Delta, used for orthonormal bases in finite or countable settings. The Fourier Transform of $\delta$ is the constant function $1$ , and the Fourier transform of $1$ is $2\pi\delta$ — making $\delta$ central to spectral analysis. In probability, $\delta$ is the density of a Dirac Measure (a point mass), connecting it to measure theory. It is the identity element of Convolution, and its rigorous home is the theory of Distributions (Generalized Functions).

🔬 Where this appears

⏳ History

- 1822 — Fourier: Joseph Fourier writes integral representations equivalent to

\delta

while studying heat conduction, implicitly using the sifting idea.

- 1828 — Green: George Green introduces what are now called Green's functions in his essay on electricity and magnetism, foreshadowing point-source methods.

- 1880s — Kirchhoff & Heaviside: Oliver Heaviside develops operational calculus with step and impulse-like objects, used boldly and without rigor in telegraphy.

- 1927 — Dirac: Paul Dirac formally introduces

\delta(x)

in The Physical Interpretation of the Quantum Dynamics, calling it an "improper function."

- 1930 — Dirac's textbook: The Principles of Quantum Mechanics spreads the delta function throughout physics.

- 1935 — Bochner: Salomon Bochner develops generalized Fourier analysis, anticipating elements of distribution theory.

- 1936 — Sobolev: Sergei Sobolev introduces generalized functions and weak solutions in the study of PDEs.

- 1945–1950 — Schwartz: Laurent Schwartz publishes the rigorous théorie des distributions, giving

\delta

a firm foundation; awarded the Fields Medal in 1950.

- 1955 — Gel'fand & Shilov: Their multivolume Generalized Functions systematizes and popularizes the theory for applied mathematics and physics.

open-problem

The delta function exposes the boundary of what classical analysis permits, and several subtleties persist:

No multiplication of distributions: There is no consistent general way to define $\delta(x)\cdot\delta(x)$ or products of distributions with overlapping singularities (Schwartz's impossibility result). This is a genuine obstacle in nonlinear PDEs and quantum field theory, where it appears as ultraviolet divergences requiring renormalization.
Choice of nascent sequence: While any unit-area family converging weakly gives the same $\delta$ , regularizing nonlinear expressions can depend on the chosen sequence (the Colombeau algebras were developed to handle such ambiguities).
Higher dimensions and curved supports: Defining $\delta$ on surfaces, manifolds, or along curves (the "single-layer/double-layer" potentials) requires careful measure-theoretic and differential-geometric setup.
Physical regularization: In practice every "point" source has finite size; whether $\delta$ -idealizations introduce spurious infinities (e.g. the self-energy of a point charge) remains a conceptual issue at the physics frontier.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also