Appendix l.A — The Dirac Delta Function

Consider a particle moving in one-dimensional space, x. A physical event associated with the particle can often be described in terms of the probability distribution of the particle in the real axis x. The probability of finding the particle in the region, x x + dx, is denoted by p x) dx. Since a point does not have a width, it takes a special consideration to describe the state that the particle is surely at a certain point x. Prom such a consideration, the Dirac delta function 6 x — x ) is obtained. A simple way is to consider the probability function p x) given by 

Equation (1.A.3) is not a well-defined mathematical function, which must have a definite value at every point x where it is defined. Dirac called it an improper function, which has the characteristic that when it occurs as a factor in an integrand the integral has a well-defined value.  

Another useful representation of the Dirac delta function is given by 

The properties of the Dirac delta function, namely, Eqs. (f.A.4)-(l.A.6), can be obtained by substituting the function giy) into the expression for 

For a vector r in the three-dimensional space with components x, y, and x, the Dirac delta function (r — r ) is represented as 

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