# A function, eh?

Not a proper

function, in the strict sense, but very useful all the same. It is defined by its properties when integrated.

**Definition:** The Dirac delta function δ(x) satisfies (integral) δ(x) dx = 1 if the range of integration includes zero, =0 if not.

Take a moment to convince yourself that no

**R**->

**R** function satisfies this. Note that for any function continious at zero, (integral) f(x)δ(x) dx = f(0).

# What use is it?

The neatest way to convince yourself of its necessity is to think of it as a

continuum generalisation of the

Kronecker delta symbol:

(sum) f_{i}δ_{ij} = f_{j}

(integral) f(x)δ(x) dx = f(0)

As such, the notion of

orthonormal eigenstates in

Dirac's own formalism can be gerneralise from the discrete eigenvalue case to the continious case:

<i|j> = δ_{ij} becomes

<x|y> = δ(x-y)

...which you need for such continious eigenvalues such as position and momentum. This, and the above stated property, make the sum-over-all-intermediate-states rule work out as it should:

|x> = (integral) |y><y|x> dy

# Generalisation to R^{n}:

**Definition:** δ^{(n)}(**x**) = δ(x_{1})δ(x_{2})...δ(x_{n})

...and then if you integrated this over some (n)volume, the result is 1 if zero is in the volume, 0 otherwise.