Delta functions

Introduction

A delta function is an infinitely narrow, infinitely high function whose area is normalized to one. This is a little difficult to understand unless we give a model for a delta function. One model is the square function. Imagine a function whose value is 1/a over the range -a/2 < x < a/2 and 0 outside these values. This function is plotted below for three values of a.

The blue trace is for a = 2. The violet trace for a = 1/2 and the red for a = 1/10. The area under each rectangle is the same because the base length is a and the height is 1/a. Since (1/a)(a) = 1 the area is always one. Imagine we take this function in the limit that a goes to zero. The function becomes narrower and taller, but always has an area of one. In the limit that a goes to zero this is a delta function. Since the delta function d(x - x0). Note that since the delta function is centered about 0 in this instance the value of x0 is 0. In this case our delta function is d(x).

In spectroscopy, a delta function is often used to represent a matching condition. If we imagine the energy levels of a molecule as being extremely narrow then the energy of an incident photon must match exactly to the energy difference in order for a transition to occur. The Einstein relation states that E2 - E1 = DE = hn. One way to write this for infinitely narrow level widths is to use a delta function d(DE - hn).

Properties

1. The delta function is the eigenfunction of the position operator. For a free particle we can operate with the position operator x (hat). The eigenvalue equation is:

The eigenvalue is x0 the actual position of the particle. The delta function specifies that of all the possible x values only x0 is non-zero.

2. The integral properties of a delta function are as follows:

  1. the integral over d(x - x0) is equal to the function evaluated at x0.

  2. The area under the delta function is one.
  1. The value of a delta function is zero everywhere except where the argument is zero.

4. A change of argument by a factor results in multiplication by the inverse of the factor.

To see this consider the above rectangular function. The delta function is 1/a over the limits -a/2 < x < a/2. Thus, the height is 1/a and the base is a. If we multiply the height by k then it becomes k/a. This means that we should multiply the base by k. In other words since

we require that

Thus,

In spectroscopy we make use of this (non-intuitive) property of the delta function to make the equation:

 

A final point is that we can use any of the lineshape functions as a representation of a delta function. A Gaussian, Lorentzian or sin(x)/x function are all delta functions in the limit that their width goes to zero. In fact

is also a representation for a delta function.