In quantum mechanics, there is a very weird function—the Dirac function. It seems to play a significant role in many fields of physics and possesses clear physical meaning, but upon closer inspection, it feels as though it is not a function at all! What exactly is this "paradoxical" entity? Let us introduce it through a physical problem:
Imagine a uniform straight line with a mass of 1 and a length of 2l. Obviously, the density of the line is \rho = \frac{1}{2l}. Placing the midpoint of the line at the origin of the coordinate axis, we have: \rho(x) = \begin{cases} \frac{1}{2l} & (-l \leq x \leq l) \\ 0 & (x < -l, x > l) \end{cases}
Therefore, we have: \int_{-\infty}^{+\infty} \rho(x)dx = 1
This equation holds for any l. Let us consider the case where l \to 0. We denote the function at this limit as \delta(x), i.e., \delta(x) = \begin{cases} \infty & (x = 0) \\ 0 & (x \neq 0) \end{cases}
Of course, the most important thing is that it maintains: \int_{-\infty}^{+\infty} \delta(x)dx = 1
This weird \delta(x) function was first proposed by Dirac, and we call it the Dirac function.
The weirdness of the Dirac function lies in its various properties that are starkly different from ordinary functions; it expands our perspective on what a function can be. Of course, since it violates the mathematical definition of an ordinary function, strictly speaking, it is not a function, but a generalized function or a functional. However, this is of little consequence to physicists. Physicists do not require the high degree of mathematical rigor; they can "play with" mathematics freely for the sake of physical research. Moreover, many concepts in mathematics were born precisely because of the "playing" with mathematics by physicists. The generalized function originating from the Dirac function is one such example.
The Dirac function does not have a significant effect when standing alone, but its utility is revealed when it acts upon other functions. Its most important property is: \int_{-\infty}^{+\infty} f(x)\delta(x)dx = f(0)
In fact, as long as the integration interval contains the point x=0, the integral is f(0). That is to say, the Dirac function uses integration to extract the value f(0) from f(x). Additionally, it is not difficult to prove that \delta(x) also has the following properties: \begin{aligned} \delta(x) &= \delta(-x) \\ x\delta(x-x_0) &= x_0 \delta(x-x_0) \end{aligned}
Since \delta(x) does not yet have an explicit expression, in early research, we would represent it using the limits of normal functions: \begin{aligned} \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{aligned}
If these are still understandable, the most weird expression of the Dirac function is undoubtedly: \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 traditional views, its limit values at other points simply do not exist!! Yet it is indeed correct, as it is a result derived from the Fourier transform. This happens to be the most 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 expressions, but that does not yield the results we want. Since the utility of the Dirac function is reflected when it acts on specific functions, we also let its derivative act on a specific function: \begin{aligned} \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{aligned}
Where integration by parts was utilized. Similarly, it can be proven that: \int_{-\infty}^{+\infty} f(x)\delta^{(n)}(x)dx = (-1)^n f^{(n)}(0).
Similarly, we can also define two-dimensional and three-dimensional Dirac functions. The n-dimensional Dirac function is denoted as \delta(\vec{r}), defined as: \delta (\vec{r}) = \begin{cases} \infty & (\vec{r} = \vec{0}) \\ 0 & (\vec{r} \neq \vec{0}) \end{cases}, \quad \int_{-\infty}^{+\infty} \delta(\vec{r}) d\vec{r} = 1
The Dirac function originated from quantum mechanics, and it is likely used most extensively in quantum mechanics. However, it has now expanded to become an important tool for solving linear partial differential equations, primarily reflected in the Green’s function method in mathematical physics equations.
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