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Methods in Theoretical Quantum Optics$

Stephen Barnett and Paul Radmore

Print publication date: 2002

Print ISBN-13: 9780198563617

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198563617.001.0001

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(p.225) Appendix 2 THE DIRAC DELTA FUNCTION

(p.225) Appendix 2 THE DIRAC DELTA FUNCTION

Source:
Methods in Theoretical Quantum Optics
Publisher:
Oxford University Press

The Dirac delta function is not a function in the usual sense of being defined for values of its argument, but is a generalized function in that it is defined by a limiting procedure and only has meaning within an integral. It is often written and manipulated without reference to an integral but it should be remembered that results so obtained are strictly prescriptions for manipulating integrals containing delta functions.

The delta function Appendix 2 The Dirac Delta Function may be defined as the limit as Appendix 2 The Dirac Delta Function of a normalized function Appendix 2 The Dirac Delta Function of x and Appendix 2 The Dirac Delta Function which is symmetric about its main or only peak at x = 0. Such functions tend to zero in the limit Appendix 2 The Dirac Delta Function for all non-zero values of x, and hence, since the function is normalized to unit area, the value at x = 0 must tend to infinity. Examples of suitable functions are Appendix 2 The Dirac Delta Function, Appendix 2 The Dirac Delta Function. A delta function Appendix 2 The Dirac Delta Function peaked at x = a is simply obtained by replacing x by xa in the function and then taking the limit Appendix 2 The Dirac Delta Function. The integral of the delta function is, for p < q,

(A2.1) Appendix 2 The Dirac Delta Function
where the value unity arises from the normalization and the value Appendix 2 The Dirac Delta Function is obtained because only one-half of the symmetric function lies within the range of integration. The Heaviside unit step function H(xa) can then be defined using (A2.1) as
(A2.2) Appendix 2 The Dirac Delta Function
The important property of Appendix 2 The Dirac Delta Function is the so-called sifting property, which states that if f(x) is a function for which f(a) is defined then
(A2.3) Appendix 2 The Dirac Delta Function
This result may be used more generally: if f(x) is itself a generalized function then (A2.3) is meaningful if f(a) is meaningful. For example,
(A2.4) Appendix 2 The Dirac Delta Function
(p.226) within an integral over either a or b. Comparing (A2.1) with (A2.3), we see that
(A2.5) Appendix 2 The Dirac Delta Function
and hence we can write
(A2.6) Appendix 2 The Dirac Delta Function
in the sense that, as before, equality holds when either side of (A2.6) lies within an integral.

The Fourier transform of Appendix 2 The Dirac Delta Function can be calculated using the sifting property (A2.3) as

(A2.7) Appendix 2 The Dirac Delta Function
Hence a representation of Appendix 2 The Dirac Delta Function obtained from the inverse transform of Appendix 2 The Dirac Delta Function is
(A2.8) Appendix 2 The Dirac Delta Function
A simple change of variable then shows that
(A2.9) Appendix 2 The Dirac Delta Function

The extension of the definition of the delta function into two or more dimensions is achieved by considering the product of individual delta functions, one for each dimension. For example, if Appendix 2 The Dirac Delta Function and Appendix 2 The Dirac Delta Function are two vectors then we define the three-dimensional delta function Appendix 2 The Dirac Delta Function as

(A2.10) Appendix 2 The Dirac Delta Function
Two-dimensional delta functions are useful in problems involving functions of complex variables, so that for a complex variable Appendix 2 The Dirac Delta Function, we define
(A2.11) Appendix 2 The Dirac Delta Function
Considering the one-dimensional delta functions in (A2.11) to be the limit of Gaussians leads to the identification
(A2.12) Appendix 2 The Dirac Delta Function
with Appendix 2 The Dirac Delta Function (see (4.4.18)). It follows from (A2.8) and (A2.11) that a representation of Appendix 2 The Dirac Delta Function is
(A2.13) Appendix 2 The Dirac Delta Function
(p.227) where Appendix 2 The Dirac Delta Function and integration is implied over the whole of the complex α-plane.

Consider the integral

(A2.14) Appendix 2 The Dirac Delta Function
where the limit as Appendix 2 The Dirac Delta Function of Appendix 2 The Dirac Delta Function is Appendix 2 The Dirac Delta Function. Assuming the integrated term in (A2.14) is zero, as it will be for all functions f(x) for which the first integral exists, then in the limit Appendix 2 The Dirac Delta Function the right-hand side of (A2.14) becomes Appendix 2 The Dirac Delta Function, using (A2.3). This gives rise to the symbol Appendix 2 The Dirac Delta Function which we call the first derivative of the delta function, given by the limit Appendix 2 The Dirac Delta Function with the property
(A2.15) Appendix 2 The Dirac Delta Function
provided Appendix 2 The Dirac Delta Function exists. We can extend this to higher-order derivatives of Appendix 2 The Dirac Delta Function, written dmδ(x)/dxm, with the properties
(A2.16) Appendix 2 The Dirac Delta Function
An alternative representation of Appendix 2 The Dirac Delta Function can be found by considering
(A2.17) Appendix 2 The Dirac Delta Function
using (A2.15), provided g(0) exists. Further, using the sifting property (A2.3) with f(x) = −g(x), we have
(A2.18) Appendix 2 The Dirac Delta Function
Comparing (A2.17) with (A2.18) leads to the identification
(A2.19) Appendix 2 The Dirac Delta Function
with the usual understanding that this should be interpreted as a rule for manipulating integrals containing delta functions. The two-dimensional extension of (A2.16) for Appendix 2 The Dirac Delta Function and Appendix 2 The Dirac Delta Function, where Appendix 2 The Dirac Delta Function is a complex variable, is
(A2.20) Appendix 2 The Dirac Delta Function