Fourier Analysis and Distributions

Fourier analysis converts localization into spectral content, differentiation into multiplication, and convolution into products. In quantum mechanics it is also the unitary change of representation between position and momentum. Distribution theory extends these statements to plane waves, delta functions, singular kernels, Green functions, and scattering prescriptions that are not ordinary integrable functions.

The formulas are convention-sensitive. A reliable calculation begins by declaring the transform pair, keeping the transform and inverse together, and checking every factor of $2\pi$, $\hbar$, and every integration measure. This chapter uses the canonical Fourier Transform Conventions throughout.

The position–momentum convention

In one dimension, write the position-space and momentum-space representatives of one state as $\psi(x)$ and $\phi(p)$. The symmetric convention is

$$\phi(p) =\frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{-ipx/\hbar}\psi(x)\,dx,$$

and

$$\psi(x) =\frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{ipx/\hbar}\phi(p)\,dp.$$

The phase $px/\hbar$ is dimensionless, the signs are opposite, and both directions carry the same normalization. Fourier Transform owns the forward map; Inverse Fourier Transform owns reconstruction and its delta-kernel logic.

Many mathematical references use wave number $k=p/\hbar$ and

$$\widetilde\psi(k) =\frac{1}{\sqrt{2\pi}} \int e^{-ikx}\psi(x)\,dx.$$

The amplitudes are related by the measure-preserving Jacobian

$$\widetilde\psi(k) =\sqrt{\hbar}\,\phi(\hbar k), \qquad dp=\hbar\,dk.$$

Replacing $p$ by $\hbar k$ in the argument without transforming the amplitude changes normalization.

Discrete and continuous spectra

For a periodic function of period $L$, Fourier modes are discrete:

$$f(x)=\sum_{n\in\mathbb Z}c_ne^{2\pi i n x/L}, \qquad c_n=\frac1L\int_0^L f(x)e^{-2\pi i n x/L}\,dx.$$

The mode spacing is fixed by the period and boundary conditions. On the full line the spacing tends to zero and sums become integrals, with delta normalization replacing Kronecker normalization. The two constructions are related but are not interchangeable term by term without the appropriate limiting measures.

Periodic Functions and Fourier Series is the canonical discrete-mode entry. Poisson Summation Formula later reconnects lattice sums and reciprocal-lattice sums.

Unitarity and Plancherel

The Fourier transform extends from well-behaved functions to a unitary operator on $L^2(\mathbb R)$. Plancherel’s identity is

$$\int_{\mathbb R}\lvert\psi(x)\rvert^2\,dx =\int_{\mathbb R}\lvert\phi(p)\rvert^2\,dp,$$

and more generally

$$\int_{\mathbb R}\chi(x)^*\psi(x)\,dx =\int_{\mathbb R} \widehat\chi(p)^*\phi(p)\,dp.$$

Thus position and momentum wavefunctions have the same Hilbert-space norm. The integral formula may initially be defined on $L^1\cap L^2$ or on Schwartz functions; Plancherel supplies the unique $L^2$ extension. An arbitrary $L^2$ transform need not be given by an absolutely convergent pointwise integral.

Plancherel and Parseval Theorems treats both the continuous norm identity and its Fourier-series counterpart.

The operator dictionary

Assume sufficient decay and regularity for integration by parts. Under the stated convention,

$$\begin{aligned} \mathcal F_x\!\left[\frac{d\psi}{dx}\right](p) &=\frac{ip}{\hbar}\phi(p),\\ \mathcal F_x[x\psi](p) &=i\hbar\frac{d\phi}{dp},\\ \mathcal F_x\!\left[-i\hbar\frac{d\psi}{dx}\right](p) &=p\phi(p). \end{aligned}$$

This gives the momentum-space representation

$$\hat x=i\hbar\frac{d}{dp}, \qquad \hat p=p.$$

Boundary terms, domains, and distributional extensions must be checked when the stated regularity fails. Momentum Representation develops the operator actions and examples; Position and Momentum Representations owns the abstract representation change.

Products and convolution

For

$$(f*g)(x)=\int_{\mathbb R}f(x-y)g(y)\,dy,$$

the symmetric position–momentum convention gives

$$\mathcal F[f*g](p) =\sqrt{2\pi\hbar}\, \mathcal F[f](p)\mathcal F[g](p),$$

while

$$\mathcal F[fg](p) =\frac{1}{\sqrt{2\pi\hbar}} \left(\mathcal F[f]*\mathcal F[g]\right)(p).$$

The factors change with convention. Convolution organizes translation-invariant response, smearing, Green kernels, and momentum-space products. Hypotheses also matter: ordinary $L^1$ convolution, $L^2$ extensions, and distributional convolution do not have identical domains of definition. Use Convolution for the precise working rules.

Wave packets and localization

A plane wave has sharp momentum but is not normalizable on the full line. A wave packet is a normalizable superposition

$$\psi(x) =\frac{1}{\sqrt{2\pi\hbar}} \int e^{ipx/\hbar}\phi(p)\,dp$$

with square-integrable momentum amplitude. Concentrating $\phi(p)$ narrows momentum while broadening the spatial packet. Gaussian packets make the reciprocal widths explicit and can saturate the position–momentum uncertainty bound.

Wave Packets owns localization and spreading. The uncertainty theorem and its physical meaning remain canonical in Position–Momentum Uncertainty.

From functions to distributions

Let $\mathcal S(\mathbb R)$ be the Schwartz space of smooth functions whose derivatives decay faster than any inverse power. A tempered distribution is a continuous linear functional on $\mathcal S$ and belongs to $\mathcal S'(\mathbb R)$. The Fourier transform acts naturally on $\mathcal S$ and extends to $\mathcal S'$ by duality.

The delta distribution is defined by

$$\langle\delta,\varphi\rangle=\varphi(0).$$

It is not an infinitely tall ordinary function. Its change-of-variables rule is

$$\delta(g(x)) =\sum_i \frac{\delta(x-x_i)} {\lvert g'(x_i)\rvert},$$

for simple zeros $g(x_i)=0$. Plane-wave orthogonality, continuous basis resolutions, Green-function sources, and idealized point interactions all use such identities distributionally.

Read Delta Function for the core identities, then Distributions for the test-function framework.

Derivatives and singular prescriptions

The derivative of a distribution $T$ is defined by moving the derivative onto the test function:

$$\langle T',\varphi\rangle =-\langle T,\varphi'\rangle.$$

This definition makes the derivative of the Heaviside step distribution equal to $\delta$ and converts jumps in ordinary functions into delta terms. Distributional Derivatives develops this calculus and its boundary-condition applications.

The Cauchy principal value defines a symmetric limiting prescription around a singularity. The central identity is

$$\frac{1}{x\pm i0} =\operatorname{PV}\frac1x \mp i\pi\delta(x).$$

The sign follows from the denominator as written. This Sokhotski–Plemelj relation separates dispersive principal-value terms from delta-supported contributions in resolvents and scattering. Principal Value Distributions is the canonical home for the distributional statement; contour deformations remain in Contour Integration.

Lattices and Poisson summation

The Dirac-comb identity

$$\sum_{n\in\mathbb Z}\delta(x-nL) =\frac1L \sum_{m\in\mathbb Z} e^{2\pi i m x/L}$$

is a distributional form of Poisson summation. It expresses the reciprocal relation between a real-space lattice of period $L$ and wave numbers $2\pi m/L$. Applications include periodic boxes, reciprocal lattices, Bloch theory, image sums, and the bridge between Fourier series and transforms. Convergence hypotheses or distributional interpretation must accompany pointwise-looking versions of the formula.

Page map

Page	Central question
Periodic Functions and Fourier Series	How are periodic functions expanded in discrete orthogonal modes?
Fourier Transform	How is a position-space function decomposed into momentum components?
Inverse Fourier Transform	Under which convention and hypotheses is the original function reconstructed?
Plancherel and Parseval Theorems	Why are norms and inner products preserved?
Convolution	When does a sliding integral become multiplication after transformation?
Wave Packets	How do normalizable superpositions balance spatial and momentum localization?
Momentum Representation	How do states and operators look in momentum space?
Delta Function	How is ideal localization defined under an integral?
Distributions	Which test-function framework makes generalized functions precise?
Distributional Derivatives	How do derivatives encode jumps and point sources?
Principal Value Distributions	How are singular integrals and $i0$ prescriptions interpreted?
Poisson Summation Formula	How are real-space and reciprocal-space lattice sums related?
Fourier Transform Tables for QM	How are transform pairs translated safely into the local convention?

A convention audit

Before using a transform identity or table entry:

1. Write both the forward and inverse transform.
2. Record whether the spectral variable is $p$, $k$, angular frequency, or ordinary frequency.
3. Check the sign in each exponential and the placement of $2\pi$ and $\hbar$.
4. Transform the integration measure and amplitude together.
5. Verify the identity transform or a Gaussian test pair.
6. Decide whether the formula is pointwise, in $L^2$, or distributional.
7. For sampled data, move to Fast Fourier Transform and state grid, ordering, aliasing, and normalization conventions.

Canonical boundaries

• The explanatory table guide is Fourier Transform Tables for QM; the full lookup table remains in the Fourier Transform Reference.
• The mathematical transform between $x$ and $p$ lives here; state representations and operator meaning live in the Hilbert-space and Core Formalism volumes.
• Distributional kernels live here; physical propagators, resolvents, and scattering prescriptions live in Dynamics and Formulations or Approximation and Scattering.
• Continuous transforms live here; finite sampled DFT and FFT algorithms live in Numerical Mathematics.

Common mistakes

Mistake	Correction
Mixing a forward transform from one convention with an inverse from another	write the pair together before calculating
Replacing $p$ by $\hbar k$ without changing amplitude or measure	include $dp=\hbar\,dk$ and the square-root Jacobian
Treating Plancherel as a claim of pointwise convergence	it is fundamentally an $L^2$ unitary statement
Dropping boundary terms in transform derivative rules	state decay, regularity, periodicity, or a distributional interpretation
Calling a plane wave a normalized full-line state	use a wave packet or generalized-eigenvector interpretation
Treating $\delta(x)$ as an ordinary function	define it through action on test functions
Multiplying or convolving arbitrary distributions freely	verify that the requested distributional operation is defined
Copying the sign of an $i0$ identity from memory	derive it from the denominator and contour prescription

Exercises

1. Transforming the momentum operator

Assume $\psi$ and its derivative decay sufficiently fast. Derive the transform of $-i\hbar\,d\psi/dx$ under the chapter convention.

Solution

Integration by parts gives

$$\begin{aligned} \mathcal F[-i\hbar\psi'](p) &=\frac{-i\hbar}{\sqrt{2\pi\hbar}} \int e^{-ipx/\hbar}\psi'(x)\,dx\\ &=\frac{-i\hbar}{\sqrt{2\pi\hbar}} \frac{ip}{\hbar} \int e^{-ipx/\hbar}\psi(x)\,dx\\ &=p\phi(p). \end{aligned}$$

The endpoint term vanishes by the stated decay hypothesis.

2. Translating between momentum and wave number

Starting from probability normalization, derive $\widetilde\psi(k)=\sqrt{\hbar}\,\phi(\hbar k)$.

Solution

Set $p=\hbar k$, so $dp=\hbar\,dk$. Requiring

$$\lvert\phi(p)\rvert^2dp =\lvert\widetilde\psi(k)\rvert^2dk$$

gives $\lvert\widetilde\psi(k)\rvert^2=\hbar\lvert\phi(\hbar k)\rvert^2$. The transform definitions fix the same positive square-root factor and phase convention, yielding

$$\widetilde\psi(k)=\sqrt{\hbar}\,\phi(\hbar k).$$

3. Delta under a linear change of variable

For real $a\ne0$, show that

$$\delta(ax-b) =\frac1{\lvert a\rvert} \delta\!\left(x-\frac ba\right).$$

Solution

For a test function $\varphi$, substitute $u=ax-b$. The orientation of the substitution is absorbed by the absolute value:

$$\int\delta(ax-b)\varphi(x)\,dx =\frac1{\lvert a\rvert} \varphi\!\left(\frac ba\right).$$

This is exactly the action of $\lvert a\rvert^{-1}\delta(x-b/a)$.

4. Derivative of the step distribution

Let $H(x)=0$ for $x\lt0$ and $H(x)=1$ for $x\gt0$. Show that $H'=\delta$ as distributions.

Solution

For a Schwartz test function $\varphi$,

$$\langle H',\varphi\rangle =-\langle H,\varphi'\rangle =-\int_0^\infty\varphi'(x)\,dx =\varphi(0),$$

because $\varphi(x)\to0$ as $x\to\infty$. Since $\langle\delta,\varphi\rangle=\varphi(0)$, the two distributions are equal.

References

• R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., McGraw-Hill, 2000.
• G. B. Folland, Fourier Analysis and Its Applications, American Mathematical Society, 1992.
• M. J. Lighthill, Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, 1958.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume II: Fourier Analysis, Self-Adjointness, Academic Press, 1975.
• E. M. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003.
• R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, World Scientific, 2003.