The Strange Dirac Function

诡异的Dirac函数 苏剑林 · 2013-01-14 · Mathematics quantum mechanics integral function

Quantum mechanics has a very strange function—the Dirac function. It seems to play a major role in many areas of physics, and it clearly has physical meaning, yet when you look at it closely, you feel it isn't really a function at all! What exactly is this "seemingly-but-not-quite" thing? Let's introduce it through a physics problem:

Imagine a uniform straight rod of mass 1 and length $2l$. Clearly, the density of the rod is $\rho=\frac{1}{2l}$. If we place the midpoint of the rod at the origin of the coordinate axis, then we have

$$\rho(x)=\left\{ \begin{array}{c}\frac{1}{2l} (-l \leq x \leq l)\\0 (x < -l , x > l)\end{array}\right.$$

so that

$$\int_{-\infty}^{+\infty} \rho(x)dx=1$$

This expression holds for any $l$. Let's consider the case $l \to 0$, and denote the function in this case by $\delta(x)$, that is

$$\delta(x)=\left\{ \begin{array}{c} \infty (x = 0) \\ 0 (x \neq 0) \end{array}\right.$$

Of course, the most important thing is that it preserves:

$$\int_{-\infty}^{+\infty} \delta(x)dx=1$$

This strange $\delta(x)$ function was first proposed by Dirac, so we call it the Dirac function.

The strangeness of the Dirac function lies in its various properties, which are completely different from those of ordinary functions—it broadens our view of what a "function" can be. Of course, since it violates the mathematical definition of an ordinary function, strictly speaking it is not a function at all, but rather a kind of generalized function or functional. But this is beside the point for physicists—physicists don't need mathematical rigor in the strict sense; they can freely "play around" with mathematics for the sake of physical research, and indeed, quite a few mathematical concepts were born precisely from physicists' "playing around" with mathematics. The generalized functions arising from the Dirac function are one such example.

On its own, the Dirac function doesn't seem to do much of anything obvious, but when it acts on other functions, its role becomes apparent. The most important property is:

$$\int_{-\infty}^{+\infty} f(x)\delta(x)dx=f(0)$$

In fact, as long as the interval of integration contains the point x=0, the integral evaluates to $f(0)$. In other words, the Dirac function uses integration to extract the value $f(0)$ from $f(x)$. It's also easy to show that $\delta(x)$ has the following property:

$$\begin{eqnarray*} \delta(x)=\delta(-x) \\ x\delta(x-x_0)=x_0 \delta(x-x_0)\end{eqnarray*} $$

Since $\delta(x)$ doesn't have an explicit expression, in the early days of studying it, people would represent it as a limit of ordinary functions:

$$\begin{eqnarray*} \delta(x)=\lim_{a\to \infty} \sqrt{\frac{a}{\pi}}e^{-ax^2} \\ \delta(x)=\frac{1}{\pi} \lim_{a \to 0}\frac{a}{x^2+a^2}\end{eqnarray*} $$

If these are still understandable, then the most bizarre expression for the Dirac function has to be:

$$\delta(x)=\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega x}d\omega$$

Although it is clearly infinite at x=0, according to conventional wisdom, its limiting value at every other point simply doesn't exist!! And yet it is indeed correct—this result follows from the Fourier transform. This happens to be the primary form in which the Dirac function appears in mathematical physics equations!

We can also define the derivative of $\delta(x)$. You might think of differentiating its limit expression, but that won't get us what we want. Since the effect of the Dirac function shows up when it acts on a concrete function, we likewise define its derivative by having it act on a concrete function:

$$\begin{eqnarray*} \int_{-\infty}^{+\infty} f(x)\delta'(x)dx=\int_{-\infty}^{+\infty} f(x)d\delta(x) \\ =[f(x)\delta(x)]|_{-\infty}^{+\infty}-\int_{-\infty}^{+\infty} f'(x)\delta(x)dx \\ =-f'(0)\end{eqnarray*} $$

where integration by parts is used. By the same reasoning, one can show: $\int_{-\infty}^{+\infty} f(x)\delta^{(n)}(x)dx=(-1)^n f^{(n)}(0)$

Similarly, we can define two-dimensional and three-dimensional Dirac functions; the n-dimensional Dirac function is denoted $\delta(\vec{r})$. It is defined as

$$\delta (\vec{r})=\left\{\begin{array}{c}\infty (\vec{r} =\vec{0})\\0 (\vec{r}\neq \vec{0})\end{array}\right.,\int_{-\infty}^{+\infty} \delta(\vec{r}) d\vec{r}=1$$

The Dirac function originated in quantum mechanics, and it is probably still used most heavily there. But it has since grown into an important tool for solving linear partial differential equations, primarily through the Green's function method in mathematical physics equations.