Distributions

A distribution is a generalized function defined by how it acts on test functions. The idea is not that singular objects become ordinary functions. The idea is that many singular expressions used in quantum mechanics become precise when they are always placed under a pairing with a sufficiently nice test function.

The delta function, plane waves, point sources, Green-function kernels, and many Fourier-transform identities are best understood this way.

Test Functions

Test functions are smooth functions chosen to make pairings, integrations by parts, limits, and Fourier transforms well behaved. Two common choices are:

• $C_c^\infty(\mathbb R)$: smooth functions with compact support;
• $\mathcal S(\mathbb R)$: Schwartz functions, which are smooth and rapidly decreasing together with all derivatives.

For Fourier analysis on the real line, $\mathcal S(\mathbb R)$ is especially convenient because the Fourier transform maps Schwartz functions to Schwartz functions.

The test function is not usually the physical state. It is the controlled input used to test a singular object.

Distributions as Linear Functionals

On a chosen test-function space $\Phi$, a distribution is a continuous linear functional

$$T:\Phi\to\mathbb C.$$

Its action on a test function $\varphi$ is often written as a pairing:

$$\langle T,\varphi\rangle.$$

This pairing is not a Hilbert-space inner product. It is notation for applying the distribution $T$ to the test function $\varphi$.

Linearity means

$$\langle T,a\varphi+b\chi\rangle = a\langle T,\varphi\rangle + b\langle T,\chi\rangle.$$

Continuity means that if test functions converge in the test-function topology, their values under $T$ converge as complex numbers. This condition is what makes distributional limits behave reliably.

Ordinary Functions as Distributions

Many ordinary functions define distributions. If $f$ is locally integrable, define

$$\langle T_f,\varphi\rangle = \int_{-\infty}^{\infty} f(x)\varphi(x)\,dx.$$

This embeds a large class of ordinary functions into the distributional framework. The notation is useful because it lets ordinary functions and singular objects be handled by the same testing rule.

For example, the constant function $1$ defines

$$\langle T_1,\varphi\rangle = \int_{-\infty}^{\infty} \varphi(x)\,dx,$$

which is perfectly meaningful for compactly supported or Schwartz test functions.

Delta Distribution

The delta distribution at $a$ is the evaluation functional

$$\langle \delta_a,\varphi\rangle = \varphi(a).$$

In integral notation this is written

$$\int_{-\infty}^{\infty} \delta(x-a)\varphi(x)\,dx = \varphi(a),$$

but the integral expression is notation for the distributional action. The delta distribution is not a function with a literal infinite value at one point.

A useful delta sequence is

$$\delta_\epsilon(x) = \frac{1}{\sqrt{\pi}\,\epsilon} e^{-x^2/\epsilon^2}, \qquad \epsilon>0.$$

For every Schwartz test function,

$$\lim_{\epsilon\to0^+} \int_{-\infty}^{\infty} \delta_\epsilon(x)\varphi(x)\,dx = \varphi(0).$$

Thus $\delta_\epsilon\to\delta$ distributionally, not pointwise as ordinary functions.

Plane Waves

A plane wave on the full line is not square-integrable:

$$\int_{-\infty}^{\infty} \lvert e^{ipx/\hbar}\rvert^2\,dx = \infty.$$

But it defines a tempered distribution on $\mathcal S(\mathbb R)$ by

$$\langle T_p,\varphi\rangle = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{-ipx/\hbar}\varphi(x)\,dx.$$

This is the rigorous shadow of the formal momentum bra $\langle p\vert$. The corresponding ket notation is useful, but it should not be confused with a normalizable Hilbert-space vector.

For the larger Hilbert-space setting that contains generalized kets, see Rigged Hilbert Spaces, First Look.

For the practical notation of $\lvert x\rangle$ and $\lvert p\rangle$, see Generalized Eigenvectors.

Fourier Transforms of Distributions

Tempered distributions can be Fourier transformed by testing against transformed test functions. In practice, this makes familiar formal identities meaningful.

Using the ordinary $k$ convention

$$F(k) = \int_{-\infty}^{\infty} f(x)e^{-ikx}\,dx,$$

one has distributional transform pairs such as

$$\mathcal F[1](k) = 2\pi\delta(k),$$

and

$$\mathcal F[e^{ik_0x}](k) = 2\pi\delta(k-k_0).$$

These are not ordinary function equalities. They mean that the two sides give the same answer after pairing with any appropriate test function.

In the wavefunction convention, the same principle underlies

$$\langle x\vert p\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar},$$

and

$$\langle p\vert p'\rangle = \delta(p-p').$$

The delta normalization is a distributional normalization rule, not a finite Hilbert-space norm.

Distributional Equalities

A distributional equality

$$T=S$$

means

$$\langle T,\varphi\rangle = \langle S,\varphi\rangle$$

for every test function $\varphi$ in the chosen test space.

This is why distributional formulas should be checked under an integral or pairing. Writing

$$\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ik(x-y)}\,dk = \delta(x-y)$$

is shorthand for a statement about what happens when the kernel is integrated against a test function.

Why Physics Needs Distributions

Quantum mechanics uses distributions because many useful idealizations are singular:

• exact position and momentum eigenstates;
• delta-normalized continuous bases;
• point sources in Green-function equations;
• delta potentials and contact interactions;
• distributional derivatives of jumps and singular kernels;
• principal-value singular integrals and $i0$ boundary-value prescriptions;
• plane-wave scattering states;
• Fourier transforms of constants, steps, and pure oscillations;
• Dirac combs and Poisson summation identities;
• boundary-value prescriptions such as outgoing or incoming waves.

The safe habit is to ask what test functions or physical wave packets make the formula meaningful. Normalizable packets usually carry the physical probability interpretation; distributions organize the limiting basis and response calculations.

What Distributions Do Not Permit

Distribution theory is powerful, but it is not permission to multiply or square singular objects arbitrarily. Products such as

$$\delta(x)^2$$

are not automatically defined in ordinary distribution theory. Some specialized frameworks assign meanings to selected products, but the definition must be stated.

Likewise, distributional convergence is not pointwise convergence. A sequence can converge to a delta distribution while becoming unbounded and narrower as ordinary functions.

Common Mistakes

• Treating $\delta(x)$ as an ordinary function with a value at $x=0$.
• Reading a distributional equality as pointwise equality.
• Forgetting to specify the test-function space when rigor matters.
• Treating plane waves as normalizable Hilbert-space states.
• Multiplying distributions as though all products were defined.
• Confusing distributional convergence with uniform, pointwise, or $L^2$ convergence.
• Using delta-normalized kets to bypass the spectral theorem or operator-domain questions.

Cross-Links

• Delta Function
• Generalized Eigenvectors
• Distributional Derivatives
• Principal Value Distributions
• Fourier Transform
• Inverse Fourier Transform
• Poisson Summation Formula
• Rigged Hilbert Spaces, First Look
• Continuous Spectra
• Real Analysis Essentials
• Green Functions
• Fourier Transform Table

References

• I. M. Gel’fand and G. E. Shilov, Generalized Functions, Volume 1, Academic Press, 1964.
• L. Schwartz, Théorie des distributions, Hermann, 1966.
• G. B. Folland, Fourier Analysis and Its Applications, American Mathematical Society, 1992.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Academic Press, 1980.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

1. Show that a locally integrable function defines a linear distribution.

Solution

Let

$$\langle T_f,\varphi\rangle = \int f(x)\varphi(x)\,dx.$$

Then

$$\begin{aligned} \langle T_f,a\varphi+b\chi\rangle &= \int f(x) \left[ a\varphi(x)+b\chi(x) \right]dx \\ &= a\int f(x)\varphi(x)\,dx + b\int f(x)\chi(x)\,dx \\ &= a\langle T_f,\varphi\rangle + b\langle T_f,\chi\rangle. \end{aligned}$$

Thus $T_f$ is linear. Continuity depends on the chosen test-function topology and the allowed class of $f$.

2. Verify that the Gaussian delta sequence has unit integral.

Solution

Compute

$$\int_{-\infty}^{\infty} \frac{1}{\sqrt{\pi}\,\epsilon} e^{-x^2/\epsilon^2}\,dx.$$

Set $u=x/\epsilon$, so $dx=\epsilon\,du$. Then

$$\frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} e^{-u^2}\,du = 1.$$

The unit integral is necessary but not by itself the whole distributional convergence statement.

3. Why can a plane wave define a distribution even though it is not in $L^2(\mathbb R)$?

Solution

The plane wave has constant magnitude, so its $L^2$ norm on the full line diverges. However, for a Schwartz test function $\varphi$,

$$\int_{-\infty}^{\infty} e^{-ipx/\hbar}\varphi(x)\,dx$$

is absolutely convergent because $\varphi$ decays rapidly. Therefore the plane wave defines a continuous linear functional on the Schwartz space, even though it is not a normalizable Hilbert-space vector.

4. Interpret $\mathcal F[1](k)=2\pi\delta(k)$ as a distributional equality.

Solution

The equality means that, for every appropriate test function $\varphi(k)$,

$$\left\langle \mathcal F[1], \varphi \right\rangle = \left\langle 2\pi\delta, \varphi \right\rangle.$$

The right side is

$$2\pi\varphi(0).$$

Thus the formula says that the Fourier transform of a constant acts on test functions by extracting their zero-frequency value, multiplied by $2\pi$ under the stated convention.