Quantum That ClicksBonus ／ Quantum Mechanics and the Fourier Transform ── a record and a CD are the same song

The math behind the "continuous wave ↔ discrete components" we met in Episodes 4 & 5 ── position and momentum are the front and back of a Fourier transform

Quantum Mechanics and the Fourier Transform A waveform (a record) and its frequency components (a CD) are two faces of the same song. Position ψ(x) and momentum φ(p) too are
the front and back of a Fourier transform, with ℏ as the exchange rate. Uncertainty falls out of this duality automatically.

Tools you'll need: superposing sine waves, Episode 1's p=ℏk, Episode 5's Δx·Δp This episode: ψ(x) and φ(p) are a Fourier-transform pair

In Episodes 4 and 5 we said that "a continuous waveform and its discrete components (a record and a CD) are two views of the same thing," and promised that the math behind it is the Fourier transform. In this bonus episode we dig straight into that. Fourier's discovery, in one line, is this: any waveform whatsoever can be built by superposing pure sine waves (waves of a single wavelength/frequency). And conversely, any waveform can be decomposed into "which sine waves are present, and how much of each." In quantum theory this carries decisive meaning ── a wave function seen "in terms of position, $\psi(x)$" and seen "in terms of momentum, $\varphi(p)$" are exactly the front and back of a Fourier transform. Two representations of one and the same state. And the exchange rate that links them is $\hbar$ ($p=\hbar k$). The intuition you had ── "these two descriptions are saying the same thing" ── takes its most precise form right here.

01Waveform and spectrum ── a record and a CD are the same song

Picture some music. The groove of a record is the waveform of the air's vibration itself (time → amplitude, continuous). A CD or a spectrum analyzer, on the other hand, holds that sound decomposed into which frequencies it contains, and how much of each. Waveform and spectrum ── opposite in form, yet carrying exactly the same song. You can go back and forth from one to the other without losing any information. The math of this round trip is the Fourier transform.

Fourier ── any waveform is a superposition of sine waves

A waveform $f(x)$ = sine waves $e^{ikx}$ of various wave numbers $k$, added up with weights $\tilde f(k)$: $f(x)=\int \tilde f(k)\,e^{ikx}\,dk$. The weight $\tilde f(k)$ is the "spectrum" ── which wavelengths are present, and how much of each. The waveform ($f$) and the spectrum ($\tilde f$) are the front and back of the same information.

02Position ψ(x) and momentum φ(p) are the front and back of a Fourier transform

In Episode 1, a wave with a definite momentum was $e^{ipx/\hbar}$ (a pure sine wave with wave number $k=p/\hbar$). Which means ── a wave with definite momentum is precisely the "pure sine wave" of quantum theory. So Fourier's language carries over word for word.

The two representations of the wave function (a Fourier-transform pair)

$$\psi(x)=\int \varphi(p)\,e^{ipx/\hbar}\,dp \qquad\Longleftrightarrow\qquad \varphi(p)=\frac{1}{2\pi\hbar}\int \psi(x)\,e^{-ipx/\hbar}\,dx$$

$\psi(x)$ (the wave function seen in terms of position = where it's likely to be) and $\varphi(p)$ (the wave function seen in terms of momentum = what momentum it's likely to have) are linked by a Fourier transform through $\hbar$ ── two representations of the same state. Neither is the "real" one ── like a record and a CD. To measure momentum is to Fourier-decompose $\psi$ and read off "which sine waves are present, and how much of each."

This is also the deeper meaning of Episode 2's $\psi=|\psi|e^{iS/\hbar}$. A wave whose phase $S/\hbar$ turns like $px/\hbar$ through space ── that is the "sine wave" of momentum $p$. The coefficients of decomposing $\psi(x)$ into such sine waves are $\varphi(p)$.

03Uncertainty is a Fourier theorem itself

Fourier has an unshakable theorem ── a narrow waveform is made of a wide spectrum (the bandwidth theorem). To make an instantaneous pulse, you have to blend in many frequencies. Conversely, a wave of just a single frequency spreads out over all of time and space. Put in symbols, between the spread of the waveform $\Delta x$ and the spread of the spectrum $\Delta k$ ──

Fourier's bandwidth theorem → the uncertainty principle $$\Delta x\cdot\Delta k\ge\tfrac12\quad\xrightarrow{\ p=\hbar k\ }\quad \Delta x\cdot\Delta p\ge\frac{\hbar}{2}$$

The left side is a purely wave (signal-processing) theorem with nothing quantum about it. It holds for sound and for radio waves. Just multiply it by Episode 1's exchange rate $p=\hbar k$ and out comes the uncertainty principle of Episode 5. So uncertainty is not a "strange assumption" of quantum theory ── it is the ordinary, inevitable consequence of Fourier that comes along the moment you represent a wave two ways. $\hbar$ merely puts a physical scale on that fact about waves.

04Try it ── Fourier synthesis: a narrow spectrum is a wide wave

The figure below. Top is the waveform $\psi(x)$; bottom is the spectrum (which wave numbers $k=p/\hbar$ are present, and how much of each). With the slider you can change the width $\Delta k$ of the sine waves being blended (how broad the spectrum is).

Make the spectrum narrow (only a few wave numbers) and the waveform on top becomes a long, spread-out wave train (momentum is sharp, position is fuzzy). Broaden the spectrum (blend in many wave numbers) and the waveform becomes a sharp pulse collected at one point (position is sharp, momentum is fuzzy). To make a narrow pulse you need many sine waves ── this is Fourier, and it is uncertainty. The product of the top and bottom spreads $\Delta x\cdot\Delta k$ cannot drop below $1/2$.

Figure: top = waveform ψ(x) (a superposition of many sine waves), bottom = spectrum (the distribution of wave numbers k=p/ℏ present). Broaden Δk and the waveform becomes a sharp pulse; narrow it and it becomes a long wave train. A narrow pulse = many sine waves (uncertainty)

Spectrum width Δk (breadth of blended wave numbers) 0.8

waveform ψ(x) spectrum φ(k)

05Time and energy, and Fourier all around us

Exactly the same duality holds between time and energy. Since $E=\hbar\omega$ (Episode 1), a time waveform $\psi(t)$ and an energy spectrum are also a Fourier pair through $\hbar$. So ──

Lifetime and linewidth ── a short-lived state has a fuzzy energy A state with a precisely defined energy keeps up the same vibration forever as $e^{-iEt/\hbar}$ ── a pure tone (Episode 4's stationary state). Conversely, a state that decays with lifetime $\tau$ is a "short tone" cut off in time, so its energy is blurred by a width $\Delta E\sim\hbar/\tau$. The reason the spectral lines of atoms and particles have a width is exactly this (the natural linewidth) ── the shorter-lived a particle, the fuzzier the measured value of its energy (mass). Episode 5's $\Delta t\cdot\Delta E\ge\hbar/2$ is this very time–energy Fourier duality.

Fourier is not a tool for quantum theory alone. Diffraction gratings and prisms are physical Fourier transformers that split light into its wavelengths; electron diffraction and X-ray crystallography are Fourier decompositions of matter waves; MRI and audio/image compression (JPEG, MP3) are all Fourier too. And the sampling of Episode 5 of "Cosmology That Clicks" (the sampling theorem that links continuous and discrete) is the flip side of Fourier as well. The single point that you can move freely between the continuous (record) and its components (CD) runs through everything from signal processing to quantum mechanics ── the "two descriptions are saying the same thing" you felt was, it turns out, the backbone of physics and mathematics that goes by the name Fourier duality.

◇ ◇ ◇

The honest line ── the reach of Fourier duality

That the position representation $\psi(x)$ and the momentum representation $\varphi(p)$ are a Fourier-transform pair with $\hbar$ at the core; that momentum eigenstates are plane waves $e^{ipx/\hbar}$; that the uncertainty $\Delta x\,\Delta p\ge\hbar/2$ follows from the bandwidth theorem $\Delta x\,\Delta k\ge1/2$ and $p=\hbar k$; the time–energy Fourier duality and the natural linewidth $\Delta E\sim\hbar/\tau$; and that diffraction, crystallography, and signal processing are physical Fourier ── these are all established physics and mathematics.

A few notes. ① Position and momentum form a Fourier pair because they are canonically conjugate. Not every pair of observables works this way (see Episode 5's discussion of conjugate pairs). ② Strictly, a plane wave $e^{ipx/\hbar}$ is an idealized state that cannot be normalized; real states are treated as wave packets (which can be normalized). ③ "CD/record" is an intuitive analogy for continuous ↔ discrete; the Fourier transform (a continuous spectrum) and sampling / Fourier series (discrete) are related but distinct ── the sampling theorem bridges them. ④ This duality is within the framework of non-relativistic quantum mechanics; in quantum field theory it is extended to the Fourier expansion of field operators (creation and annihilation operators).

Practice problems (solvable with just this episode's ideas)

A state with a precisely defined momentum ── what kind of wave is it in position, and why is its position fuzzy?
See the answer
A plane wave $e^{ipx/\hbar}$ of a single wave number $k=p/\hbar$. It spreads uniformly over all of space, so its position is completely undetermined (in Fourier, a single frequency = an infinitely spread-out wave).
To make a sharp pulse in position, what has to happen to the momentum (wave number) components?
See the answer
You have to blend in many wave numbers (a wide spectrum). Narrow waveform = wide spectrum (the bandwidth theorem). So sharpening the position blurs the momentum = uncertainty.
Why does a particle with a shorter lifetime τ have a fuzzier measured energy (mass)?
See the answer
The time waveform and energy are a Fourier pair. Short-lived = a short "tone" in time = a wide energy width ΔE∼ℏ/τ (the natural linewidth). This is why the mass of a short-lived particle has a width.
Separate the "quantum part" and the "purely wave part" of the uncertainty Δx·Δp≥ℏ/2.
See the answer
The purely wave (Fourier) part: Δx·Δk≥1/2 (which holds in signal processing too). The quantum part: p=ℏk (exchanging wave number for momentum via ℏ). Multiply them and you get uncertainty. ℏ gives a physical scale to a fact about waves.

Bonus summaryPosition and momentum are the front and back of a Fourier transform, joined by ℏ

Any waveform can be built from a superposition of pure sine waves (Fourier). In quantum theory, the wave of definite momentum $e^{ipx/\hbar}$ is that "sine wave," and the position representation $\psi(x)$ and the momentum representation $\varphi(p)$ are a Fourier-transform pair with $\hbar$ at the core ── two faces of the same state (a record and a CD). To measure momentum is to Fourier-decompose $\psi$. And multiply the bandwidth theorem "a narrow waveform is a wide spectrum" $\Delta x\Delta k\ge1/2$ by $p=\hbar k$ and out comes uncertainty $\Delta x\Delta p\ge\hbar/2$, automatically ── uncertainty is not a strange assumption but the inevitable consequence of representing a wave two ways.

The same duality holds for time and energy ($E=\hbar\omega$), giving the natural linewidth by which a short-lived state's energy blurs. From diffraction and crystallography to MP3 ── and all the way to the sampling of "Cosmology That Clicks" ── the Fourier duality that lets you move freely between the continuous (waveform) and its components (spectrum) runs through it all. The "two description systems are saying the same thing" you felt ── its true identity was Fourier duality, the backbone of physics and mathematics. The dimensionful $x,p$ are stage props; what links them is $\hbar$, the scale of a ratio.

Toward the collection's capstone With this bonus, quantum theory's "the two descriptions are the same" has been made precise as Fourier duality. Seen more broadly ── relativity and quantum theory, fields and gravity, all connect into a single diagram along the axes of the fundamental constants $c,\hbar,G$. As the closing of the collection, we've prepared that map separately: "The Cube of Physics" (which corner each series sits at). Drop the units, look from ratios and axes, and physics fits on a single page.

This document is the bonus episode of the "Quantum That Clicks" series, a reader for physics-loving high-schoolers and undergraduates. That the position representation and the momentum representation are linked by a Fourier-transform pair (kernel $e^{\pm ipx/\hbar}$); that the momentum eigenfunctions are plane waves; the uncertainty $\Delta x\,\Delta p\ge\hbar/2$ from the bandwidth theorem $\Delta x\,\Delta k\ge1/2$ and $p=\hbar k$; the time–energy Fourier duality and the natural linewidth $\Delta E\approx\hbar/\tau$; and that diffraction, crystallography, and signal processing are physical Fourier analysis ── these are all established physics and mathematics. That position and momentum form a Fourier pair because of canonical conjugacy and not every pair of observables does; that a plane wave is a non-normalizable idealized state while real states are wave packets; that the continuous Fourier transform and sampling / Fourier series (discrete) are related but distinct concepts bridged by the sampling theorem; and that this document is within the framework of non-relativistic quantum mechanics while in quantum field theory it extends to the Fourier expansion of fields (creation and annihilation operators) ── these are stated plainly in the "honest line" in the body. The figure is a schematic of the Fourier synthesis of a Gaussian wave packet (with $\Delta x\cdot\Delta k$ constant). ── To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and the answers are frozen and hidden). Related: Episode 5, Uncertainty ／ Contents ／ The Cube of Physics (collection).

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, changing the spectrum width Δk turns the waveform between a wave train and a pulse. Click "See the answer" to open a solution.