The Dirac Delta

In Chapter 3 (Section 15) of his 1930 book AThe Principles of Quantum Mechanics@, Paul Adrien Maurice Dirac (1902 Aug 08 - 1984 Oct 20) defined and described what he called an improper function that he had devised as a means of simplifying some of the calculations that he had to carry out in his version of quantum mechanics. We now call that improper function the Dirac Delta and I can offer no better description of it than to quote Dirac directly:

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AOur work in ' 10 led us to consider quantities involving a certain kind of infinity. To get a precise notation for dealing with these infinities, we introduce a quantity δ(x) depending upon a parameter x satisfying the condition

(Eq=n 1)

To get a picture of
δ(x),
take a function of the real variable x which vanishes everywhere except inside a
small domain, of length ε
say, surrounding the origin x=0, and which is so large inside this domain that
its integral over this domain is unity. The exact shape of the function inside
this domain does not matter, provided there are no unnecessarily wild variations
(for example provided the function is always of order
ε^{-1}).
Then in the limit ε6
0 this function will go over into
δ(x).@

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Fudging on Dirac=s Avanishes everywhere except@ criterion, we can take the formula of the normal (or Gaussian) distribution, the famous bell-shaped curve, as our function,

(Eq=n 2)

As we diminish the value of a the curve narrows into a spike centered on the origin of the coordinate axis. Taking the limit, we get

(Eq=n 3)

Many other functions could take the place of the Gaussian in Equation 2, but Dirac cautions us to remain aware that the limit process removes something fundamental from the function:

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Aδ(x) is not a function of x according to the usual mathematical definition of a function, which requires a function to have a definite value for each point in its domain, but is something more general, which we may call an >improper function= to show up its difference from a function defined by the usual definition. Thus δ(x) is not a quantity which can be generally used in mathematical analysis like an ordinary function, but its use must be confined to certain simple types of expression for which it is obvious that no inconsistency can arise.@

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So what good can we make of this function from which we have removed the property of being a function? Dirac tells us that AThe most important property of δ(x) is exemplified by the following equation

(Eq=n 4)

where f(x) is any continuous function of x.@ You may feel, as I did, a temptation to prove that equation using integration by parts with Equation 1: it doesn=t work (improper function, indeed!) and it=s unnecessary. Dirac defined his delta to be such a narrow spike that over the domain where it has nonzero value the function f(x) can be taken as a constant with value f(0). Treating the function as if it were constant allows us to process Equation 4 as

(Eq=n 5)

By reducing a function to an effective constant in that way Dirac has applied an idea similar to the one underlying integration itself, that of dividing the area under a curve into extremely narrow trapezoids and then making the approximation that the trapezoids have the areas of rectangles of the same width and median length.

We still need to take care in applying the Dirac delta. Dirac said that A...although an improper function does not itself have a well-defined value, when it occurs as a factor in an integrand the integral has a well-defined value. In quantum theory, whenever an improper function appears, it will be something which is to be used ultimately in an integrand. Therefore it should be possible to rewrite the theory in a form in which the improper functions appear all through only in integrands. One could then eliminate the improper functions altogether. The use of improper functions thus does not involve any lack of rigour in the theory, but is merely a convenient notation, enabling us to express in a concise form certain relations which we could, if necessary, rewrite in a form not involving improper functions, but only in a cumbersome way which would tend to obscure the argument.@

Having defined the delta through integration, Dirac then showed that it also serves as a derivative for discontinuous functions, which use provides an alternative definition of the delta. Take, as Dirac did, the step function

(Eq=n 6)

Dirac claims that the derivative of that function

(Eq=n 7)

He then proves and verifies that proposition through an integration by parts.

Multiply a continuous function f(x) by ds(x)/dx
and integrate the product with respect to x between the limits x=x_{1}>0
and x=x_{2}<0. We have

(Eq=n 8)

which is the same result that we obtained in Equation 5; therefore, Equation 7 stands true to mathematics. Dirac commented, AThe δ function appears whenever one differentiates a discontinuous function.@

Dirac offers the derivative of the natural logarithm as an example of what he meant by that comment. Conventionally we have

(Eq=n 9)

But we also know that

(Eq=n 10)

Thus, the proper natural logarithm includes an imaginary-valued step function that has its discontinuity at x=0. That means that we should properly write Equation 9 as

(Eq=n 11)

That particular combination of a reciprocal function and the delta function, Dirac commented, plays an important part in the quantum theory of collisions: as Dirac explained it in Section 50 of his book, the quantum description of a particle collision must include the formula on the right side of Equation 11 as a factor in order to describe only outgoing particles emerging from the collision.

Dirac also listed six equations that elaborated the properties of the delta:

aIb

(Eq=n 12)

This equation simply expresses the fact that the delta is symmetric about the point x=0, giving it the character of an even function of x.

aIIb

(Eq=n 13)

Because δ(x) has a non-zero value only at x=0, this might seem trivially true to Mathematics, but we must remind ourselves that multiplying any number, even an infinitely large one like δ(0), always yields zero as the product. When this product appears in an integrand, we can take it as a multiplication of the integrand by zero.

aIIIb

(Eq=n 14)

This comes directly from Equation 1 and the knowledge that in order to integrate a function we must multiply it by the differential of its argument. In order to have a proper integral, then, we must multiply δ(ax) by adx, so we must have Equation 1 as

(Eq=n 15)

which is the same as Please note that in those integrations I have rendered the infinite limits implicit for clarity.

aIVb

(Eq=n 16)

In this case we factor the argument of the delta on the left side of the equation and get

(Eq=n 17)

The argument of that function goes to zero at two points on the x-axis; x=a and x=-a. At those points the nonzero factor in the argument becomes, respectively, -2a and 2a, so we get

(Eq=n 18)

Applying Equations 12 and 14 to that equation gives us Equation 16.

aVb

(Eq=n 19)

We know that

(Eq=n 20)

because the delta has its nonzero value at the point x=a. If we let f(x)=δ(x-b) in that equation, then we get Equation 19.

aVIb

(Eq=n 21)

Again, this merely encodes the fact that the delta has its nonzero value at the point x=a and thus it is equivalent to Equation 20.

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In Appendix A7 of his classic 1965 text AFundamentals of Statistical and Thermal Physics@, Dr. Frederick Reif demonstrates how to convert the Kronecker delta into an integral representation of Dirac=s delta, one that comes in handy in the quantum theory.

For integers n and m we define Kronecker=s delta as

(Eq=n 22)

Now consider the integral

(Eq=n 23)

With that integral we can represent the Kronecker delta as

(Eq=n 24)

We want to convert that function of a discrete variable into the equivalent function of a continuous variable, Dirac=s delta.

The values of n denote points on the number line. To convert that discrete array into a continuous x-axis we make the substitution

(Eq=n 25)

in which we intend to let the value of L approach infinity. Each successive step on the number line then becomes

(Eq=n 26)

That transformation makes the Kronecker delta conform to

(Eq=n 27)

We calculate

(Eq=n 28)

For convenience let

(Eq=n 29)

so that we have

(Eq=n 30)

and we have the limits of the integration going from to . Equation 28 then becomes

(Eq=n 31)

Now we must prove that equation and see whether it stands true to mathematics. We will have little difficulty with that task because Reif offers an elegant proof that verifies the proposition that Equation 31 truly represents the Dirac delta. We start by noting that for x…0 the integrand oscillates rapidly, so the integral equals zero and for x=0 the integrand reduces to dk and the integral becomes infinite. In order to avoid convergence ambiguities when the magnitude of k approaches infinity we multiply the integrand by exp[-γ|k|], in which the gamma represents a positive minuscule number. We thus have Equation 31 as

(Eq=n 32)

We take the limit of that formula as gamma approaches zero and get zero when x…0 and infinity when x=0, as we should. If we integrate that equation with respect to x, we get

(Eq=n 33)

as we require. Therefore, Equation 31 gives us a correct description of the Dirac delta.

Finally, we have a very special relationship involving the Dirac delta. In the theory of potentials we have

(Eq=n 34)

We know that for some vectorfield **V** we have Gauss=
s theorem as

(Eq=n 35)

So let V=L (1/r), which we find in the description of forcefields derived from static potentials. Then we have Equation 35 in the form

(Eq=n 36)

in which **r _{0}** represents the unit vector in the
radial direction. The upper case omega indicates that we have tacitly assumed
that the boundary of the enclosed volume consists of a spherical shell centered
on the origin of the coordinate grid, but the surface can have any shape and we
still get the same result as long as the volume includes the origin. Now
construct a spherical shell of minuscule radius around the origin and connect it
to our original bounding surface by a thin tube. We thus create a small
extension of the original surface that excludes the origin from the enclosed
volume. In the case of the surface integral over the small surface, the area
vector must point toward the origin in order to point out of the volume of which
it is part of the boundary, so the integral must take a negative sign. We thus
find that if the surface encloses a volume that excludes the origin, we have

(Eq=n 37)

and if the volume includes the origin, we have

(Eq=n 38)

which verifies Equation 34.

References

Arfken, George; Mathematical Methods for Physicists;1966, Academic Press, New York, LCCCN 65-27740.

Dirac, Paul A.M.; The Principles of Quantum Mechanics, 1930, 4^{th}
Ed. (1958), Oxford University Press, Oxford, England, ISBN 0-19-852011-5 (1991
Printing).

Reif, Frederick; Fundamentals of Statistical and Thermal Physics; 1965, McGraw-Hill, New York, LCCCN 63-22730.

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