Dirac Delta Function: Definition, Properties and Uses
The Dirac delta function δ(x) is the mathematician's way of describing an infinitely sharp spike: it is zero everywhere except at one point, yet the total area under it equals exactly 1. Physicists use it to model idealized point objects — a point charge with all its charge at one location, an impulsive hammer blow delivered in an instant, a mass concentrated at a single point. It is not an ordinary function but a distribution: it only makes sense inside integrals, where it obeys the sifting property ∫f(x)δ(x − a)dx = f(a).
The delta function matters because point-like idealizations are everywhere in physics, and δ(x) lets you write them as honest formulas instead of words. Charge densities, Green's functions for differential equations, the normalization of quantum plane waves, and the impulse in collisions all rely on it. Once you accept that δ only lives inside integrals, it becomes one of the most practical tools in mathematical physics.
Key Concepts
- Sifting property: ∫₋∞^∞ f(x)δ(x − a)dx = f(a). The delta "picks out" the value of f at the spike location. This is the defining property.
- Normalization: ∫₋∞^∞ δ(x)dx = 1. The spike has unit weight — an infinitely tall, infinitely narrow bump with area 1.
- Scaling: δ(ax) = δ(x)/|a|. Squeezing the spike horizontally by a raises its height to keep the area equal to 1 — the formula handles it automatically.
- Delta of a function: δ(g(x)) = Σᵢ δ(x − xᵢ)/|g′(xᵢ)|, summing over the roots xᵢ of g. Each root contributes a spike weighted by the inverse slope.
- Limit representation: δ(x) = lim_{ε→0} (1/√(2πε²))e^{−x²/2ε²}, a Gaussian spike narrowing to zero width while keeping unit area.
Formulas
- Sifting: ∫ f(x)δ(x − a)dx = f(a), for any continuous f. Units: δ(x) has units of 1/(unit of x).
- Scaling: δ(ax) = δ(x)/|a|; in particular δ(−x) = δ(x) — the delta is even.
- Composition: δ(g(x)) = Σᵢ δ(x − xᵢ)/|g′(xᵢ)| over simple roots xᵢ of g.
- Fourier form: δ(x) = (1/2π)∫₋∞^∞ e^{ikx}dk, exhibiting the delta as a superposition of all frequencies.
Worked Example
Model a point charge q sitting at x = 2 m on a wire with charge density ρ(x) = qδ(x − 2). The total charge between x = 0 and x = 5 is ∫₀⁵ qδ(x − 2)dx = q × 1 = q, since the spike at x = 2 lies inside the interval. Between x = 3 and x = 5 the integral is 0 — the spike is outside. Next, evaluate ∫₋₁³ x²δ(x − 1)dx: by sifting, this equals (1)² = 1. Finally, simplify δ(x² − 4): the roots are x = ±2 with |g′| = |2x| = 4 at each root, so δ(x² − 4) = [δ(x − 2) + δ(x + 2)]/4. Then ∫₀^∞ δ(x² − 4)dx = 1/4, because only the x = 2 spike falls in [0, ∞) — a result you could never guess without the composition rule.
Common Misconceptions
- "δ(0) = ∞, so the delta is just infinity at one point." Infinity is not a value the delta takes; δ is defined solely by what it does inside integrals. Asking for δ(0) as a number misses the point.
- "Since δ is zero almost everywhere, its integral must be zero." The spike at the single point carries the entire weight 1 — that is the whole idea, and the limit-of-Gaussians picture shows the area staying 1 as the width shrinks.
- "δ(x)² makes sense." Products of distributions at the same point are undefined in the standard theory — expressions like δ²(x) have no meaning without advanced machinery.
Real-World Applications
- Point charges and masses: ρ(r) = qδ³(r − r₀) writes a point charge as a genuine density for use in Maxwell's equations.
- Impulse and Green's functions: the response of any linear system to δ(t) (the impulse response) generates the response to arbitrary forces by superposition.
- Quantum mechanics: position eigenstates satisfy ⟨x|x′⟩ = δ(x − x′), and plane waves are delta-normalized.
- Signal processing: the Dirac comb (a row of deltas) models ideal sampling of continuous signals.
The Dirac delta turns "all concentrated at one point" into a formula you can integrate, differentiate, and Fourier transform. Treat it as an instruction inside integrals — sift, scale, compose — and point-like physics becomes routine calculation.