Consider a distribution that at the initial time t = 0 has the form of a Dirac delta function (x). A delta function can he

Consider a distribution that at the initial time t = 0 has the form of a Dirac delta function δ(x). A delta function can he represented by a Fourier integral:

At later times the pulse becomes

or by use of(10),

Evaluate the integral to obtain the result (14). The method can be extended to describe the time development of any distribution given at t = 0. If the distribution is f(x, 0), then by the definition of the delta function

f(x, 0) = f dx’ f(x’, 0)δ(x – x’).

The time development of δ(x – x’)is

by (14). Thus at time t the distribution f(x, 0) has evolved to

This is a powerful general solution.

0(x,0) = dkexp(ikx) 8(x) %3D 2n