Fourier Transform Tables for QM

Fourier-transform tables are useful only after the convention is fixed. The same Gaussian, exponential decay, plane wave, or delta function can carry different factors of $2\pi$, $\hbar$, or square-root normalizations depending on the transform pair being used.

This page explains how to read transform tables in quantum-mechanical calculations. The full lookup table is Fourier Transforms in the Reference. The goal here is to make the entries usable without turning them into convention traps.

Two Conventions in Play

For mathematical $k$-space calculations, this volume uses

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

with inverse

$$f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(k)e^{ikx}\,dk.$$

For wavefunctions, the site convention is the unitary momentum transform

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

and

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

The variables are related by

$$p=\hbar k, \qquad dp=\hbar\,dk.$$

If $F(k)$ is the ordinary mathematical transform of $\psi(x)$, then

$$\phi(p) = \frac{1}{\sqrt{2\pi\hbar}} F(p/\hbar).$$

This line is the simplest way to translate most table entries into momentum-space wavefunctions.

Compact Table

Under the $k$-space convention above:

$f(x)$	$F(k)$	Use in quantum mechanics
$\delta(x-a)$	$e^{-ika}$	point source, position ket normalization
$1$	$2\pi\delta(k)$	zero wave-number distribution
$e^{ik_0x}$	$2\pi\delta(k-k_0)$	plane wave as a generalized momentum state
$e^{-\alpha x^2}$	$\sqrt{\pi/\alpha}\,e^{-k^2/(4\alpha)}$	Gaussian packet, heat kernel, oscillator integrals
$e^{-\alpha\lvert x\rvert}$	$\dfrac{2\alpha}{\alpha^2+k^2}$	exponentially localized tail
$\dfrac{1}{x^2+a^2}$	$\dfrac{\pi}{a}e^{-a\lvert k\rvert}$	rational kernel with exponential transform
$\operatorname{PV}\dfrac{1}{x}$	$-i\pi\,\operatorname{sgn}(k)$	Hilbert-transform and dispersion identities

The entries involving constants, plane waves, delta functions, and principal values are distributional. They are equalities after pairing with test functions, not pointwise equalities of ordinary functions.

Gaussian Translation

The table entry

$$e^{-\alpha x^2} \longleftrightarrow \sqrt{\frac{\pi}{\alpha}} e^{-k^2/(4\alpha)}$$

means

$$\int_{-\infty}^{\infty} e^{-\alpha x^2}e^{-ikx}\,dx = \sqrt{\frac{\pi}{\alpha}} e^{-k^2/(4\alpha)}, \qquad \operatorname{Re}\alpha>0.$$

To use this for a momentum-space wavefunction, set $k=p/\hbar$ and include the unitary prefactor:

$$\phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \sqrt{\frac{\pi}{\alpha}} \exp \left( -\frac{p^2}{4\alpha\hbar^2} \right)$$

for the unnormalized position function $\psi(x)=e^{-\alpha x^2}$.

For normalized Gaussian packets, it is often clearer to work with the width directly. If

$$\psi(x) = \frac{1}{(2\pi\sigma_x^2)^{1/4}} \exp \left[ -\frac{(x-x_0)^2}{4\sigma_x^2} +\frac{i p_0(x-x_0)}{\hbar} \right],$$

then

$$\phi(p) = \left( \frac{2\sigma_x^2}{\pi\hbar^2} \right)^{1/4} \exp \left[ -\frac{\sigma_x^2(p-p_0)^2}{\hbar^2} -\frac{i p x_0}{\hbar} \right].$$

The probability density $\lvert\phi(p)\rvert^2$ is centered at $p_0$ and has width inverse to $\sigma_x$, as expected from Wave Packets.

Plane Waves and Delta Functions

The table entry

$$e^{ik_0x} \longleftrightarrow 2\pi\delta(k-k_0)$$

is distributional. In quantum notation, the normalized plane-wave kernel is

$$\langle x\vert p_0\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ip_0x/\hbar}.$$

Transforming it to momentum space gives

$$\phi(p) = \delta(p-p_0).$$

This does not mean the plane wave is square-integrable. It means the exact momentum label is delta-normalized. The Hilbert-space state used in probability calculations should be a normalizable packet or a controlled distributional idealization.

For the distributional framework, see Distributions and Generalized Eigenvectors.

Exponential Decay

The entry

$$e^{-\alpha\lvert x\rvert} \longleftrightarrow \frac{2\alpha}{\alpha^2+k^2}, \qquad \alpha>0,$$

is useful whenever a bound-state tail or one-dimensional Green-function kernel has exponential decay. The transform is broad when the real-space decay length $1/\alpha$ is short, and narrow when the real-space function is spread out.

In momentum variables, replace $k$ by $p/\hbar$:

$$F(p/\hbar) = \frac{2\alpha}{\alpha^2+p^2/\hbar^2}.$$

If this is being used as a wavefunction transform, the extra factor $1/\sqrt{2\pi\hbar}$ still applies.

Delta Shifts and Phases

The delta entry

$$\delta(x-a) \longleftrightarrow e^{-ika}$$

is the simplest example of the translation rule: shifting a point source in position creates a phase in wave-number space.

More generally,

$$f(x-a) \longleftrightarrow e^{-ika}F(k).$$

In momentum notation this phase becomes

$$e^{-ipa/\hbar}.$$

This is why a displaced wave packet has the same momentum probability distribution as the original packet, but a different momentum-space phase.

Products, Convolutions, and Potentials

Fourier tables often list multiplication and convolution rules. With the $k$ convention,

$$(f*g)(x) \longleftrightarrow F(k)G(k),$$

while

$$f(x)g(x) \longleftrightarrow \frac{1}{2\pi}(F*G)(k).$$

This matters for momentum-space quantum mechanics. A local potential $V(x)\psi(x)$ is a product in position space, so it becomes a convolution in momentum space. The detailed operator dictionary is in Momentum Representation, and the theorem-level statement is Convolution.

How to Use a Table Safely

When using a transform table, check the following before substituting:

• the sign in the exponential;
• where the factors of $2\pi$ appear;
• whether the table uses $k$ or physical momentum $p$;
• whether the transform is unitary or asymmetric;
• whether the entry is an ordinary function identity or a distributional identity;
• whether a shift, scaling, or change of variables needs a Jacobian;
• whether the result is meant as a wavefunction or as a Green-function kernel.

Most Fourier mistakes in quantum mechanics are convention mistakes, not deep analysis mistakes.

Common Mistakes

• Treating $F(k)$ and $\phi(p)$ as the same function without the factor $p=\hbar k$.
• Forgetting that $dp=\hbar\,dk$ when comparing probability densities.
• Using a table with the opposite exponential sign.
• Applying plane-wave and delta entries as if they were square-integrable functions.
• Dropping the unitary prefactor $1/\sqrt{2\pi\hbar}$ when converting a table entry into a momentum wavefunction.
• Assuming a quoted Gaussian width is the same as $\Delta x$ without checking the exponent convention.

Cross-Links

• Fourier Transform
• Inverse Fourier Transform
• Fourier Transform Conventions
• Fourier Transform Table
• Momentum Representation
• Wave Packets
• Plancherel and Parseval Theorems
• Delta Function
• Distributions
• Principal Value Distributions

References

• G. B. Folland, Fourier Analysis and Its Applications, American Mathematical Society, 1992.
• M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2006.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.

Exercises

1. Let $F(k)$ be the ordinary $k$-space transform of $\psi(x)$. Derive the relation between $F(k)$ and the momentum-space wavefunction $\phi(p)$.

Solution

By definition,

$$F(k) = \int e^{-ikx}\psi(x)\,dx.$$

The momentum transform is

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

Set $k=p/\hbar$. Then

$$\phi(p) = \frac{1}{\sqrt{2\pi\hbar}} F(p/\hbar).$$

2. Use the table to transform $\delta(x-a)$ under the $k$ convention.

Solution

Compute directly:

$$F(k) = \int_{-\infty}^{\infty} \delta(x-a)e^{-ikx}\,dx = e^{-ika}.$$

The shifted delta distribution becomes a phase.

3. Under the $k$ convention, find the transform of $e^{-\alpha\lvert x\rvert}$ for $\alpha>0$ and explain how its width changes as $\alpha$ increases.

Solution

The table gives

$$F(k) = \frac{2\alpha}{\alpha^2+k^2}.$$

As $\alpha$ increases, the real-space function becomes more localized because the decay length $1/\alpha$ decreases. Its transform becomes broader in $k$ space, consistent with Fourier uncertainty.

4. Why is the transform of $e^{ik_0x}$ a delta distribution rather than an ordinary function?

Solution

The plane wave has constant magnitude and is not integrable on the real line, so the ordinary integral

$$\int_{-\infty}^{\infty} e^{i(k_0-k)x}\,dx$$

does not converge as an ordinary integral. Distributionally it acts as $2\pi\delta(k-k_0)$, meaning it extracts the test function value at $k_0$ after pairing.

5. A table uses the opposite convention $\int f(x)e^{+ikx}\,dx$. What changes should you expect?

Solution

The signs of shifts and modulation rules reverse. For example, with the site’s $e^{-ikx}$ convention,

$$f(x-a) \longleftrightarrow e^{-ika}F(k).$$

With an $e^{+ikx}$ convention, the phase would be $e^{+ika}$ for the corresponding forward transform. Factors of $2\pi$ may also move if the inverse convention differs.