---
{
"id": "ASD-QM-THEORY-QUANTUM-UNCERTAINTY-PROBABILITY-EN",
"path": "/en/theory/quantum/uncertainty-probability",
"title": "Uncertainty: Fourier Width Bounds for Wave Packets",
"title_en": "Uncertainty: Fourier Width Bounds for Wave Packets",
"category": "Quantum Phenomena",
"summary": "Start with an envelope’s spatial width and wave-number bandwidth, derive the Fourier standard-deviation bound step by step, and identify the phase–momentum matching needed to connect it to preparation and measurement.",
"template": "theory",
"version": "1.0.0",
"updated": "2026-10-24",
"canonical": true,
"source_tier": "canonical",
"theory_status": "EFFECTIVE MODEL",
"evidence_status": "UNTESTED",
"locale": "en",
"module_key": "quantum",
"module_navigation": [
{
"path": "/en/theory/quantum",
"label": "Overview",
"en": "Overview"
},
{
"path": "/en/theory/quantum/accompanying-field",
"label": "Accompanying Field",
"en": "Accompanying Field"
},
{
"path": "/en/theory/quantum/matter-wave",
"label": "Matter Wave",
"en": "Matter Wave"
},
{
"path": "/en/theory/quantum/effective-wave-equation",
"label": "Effective Equation",
"en": "Effective Equation"
},
{
"path": "/en/theory/quantum/interference-diffraction",
"label": "Interference",
"en": "Interference"
},
{
"path": "/en/theory/quantum/uncertainty-probability",
"label": "Uncertainty",
"en": "Uncertainty"
},
{
"path": "/en/theory/quantum/detection-statistics",
"label": "Detection",
"en": "Detection"
},
{
"path": "/en/theory/quantum/tunneling",
"label": "Tunneling",
"en": "Tunneling"
},
{
"path": "/en/tests/quantum",
"label": "Requirements",
"en": "Requirements"
}
],
"canonical_scope": "Canonical research framework · effective limits are conditional",
"source_chapter": "Chapter 5",
"source_reference": "§§3.5–3.6; (5.3.23)–(5.3.27)",
"source_version": "ASD-BOOK-ddbade997b45",
"source_sha256": "dace8ff491b4e4ae8a59b2b98e2a6c9c10e689079021ce0bc117985baba68e86",
"source_kind": "manifest",
"source_records": [
{
"id": "ASD-BOOK-ddbade997b45",
"title": "各向异性空间动力学（最终合订稿）",
"filename": "各向异性空间动力学.docx",
"format": "docx",
"size_bytes": 503564,
"sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e"
}
],
"source_manifest": "final-manuscript-source-manifest.json",
"source_manifest_url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json",
"last_reviewed": "2026-09-16",
"depends_on": [
"/en/theory/quantum/matter-wave",
"/en/theory/quantum/effective-wave-equation"
],
"related": [
"/en/theory/quantum"
],
"open_problems": [
"/en/open-problems/quantum/phase-momentum-locking",
"/en/open-problems/quantum/born-statistics"
],
"equations": [
"qm-fourier",
"qm-uncertainty",
"qm-gaussian"
],
"sources": [
{
"label": "Final combined manuscript — adopted scope documented on this page",
"kind": "final-combined-manuscript",
"url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json"
},
{
"label": "Previous source: CH5-ca737ec4e327 · 第五章-出版整理稿.docx",
"kind": "historical-source"
}
],
"outline": [
"What width does a localized wave packet have?",
"How integration by parts produces a lower bound",
"From wave number to momentum requires material matching",
"Why a Gaussian width and a slit width cannot be interchanged",
"Duration and spectrum of a finite process",
"How wave-packet structure enters measurement",
"Derivation sources"
],
"seo_title": "Uncertainty: Fourier Width Bounds for Wave Packets | Quantum Phenomena Foundation v1.0 · ASD",
"canonical_url": "https://asd-physics.org/en/theory/quantum/uncertainty-probability",
"translation_path": "/theory/quantum/uncertainty-probability",
"mathematical_review": "CONDITIONAL",
"evidence_note": "NOT YET UNIQUELY TESTED · known wave phenomena are requirements; an independent ASD source and universal event statistics remain open.",
"equation": "qm-fourier",
"translation_of": "/theory/quantum/uncertainty-probability",
"translation_version": "1.0.0",
"date_published": "2026-09-16",
"metadata_reviewed": "2026-09-22",
"section_anchors": [
"uncertainty-widths",
"uncertainty-fourier-proof",
"uncertainty-momentum",
"uncertainty-gaussian",
"uncertainty-time-frequency",
"uncertainty-measurement-scope",
"uncertainty-sources"
],
"legacy_anchors": {
"s0": "uncertainty-widths",
"s1": "uncertainty-fourier-proof",
"s2": "uncertainty-gaussian",
"s3": "uncertainty-time-frequency",
"s4": "uncertainty-measurement-scope"
},
"research_details_from": 6,
"final_source": {
"source_id": "ASD-BOOK-ddbade997b45",
"chapter": "Chapter 5",
"reference": "§§3.5–3.6; (5.3.23)–(5.3.27)",
"locators": [
"p1967",
"p1971",
"p1974",
"p1979",
"p1980"
],
"adopted_scope": "Preserved width proof and explicit first-zero identity for a rectangular time window",
"conditions": [
"Selected channel and stated initial/boundary conditions; action/energy matching retained",
"Continuum, slow-envelope and detector approximations kept at their respective levels"
],
"unresolved": [],
"migration_status": "reviewed",
"manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e",
"source_manifest": "final-manuscript-source-manifest.json",
"body_review": "updated"
},
"editorial_date": "2026-10-24",
"previous_source": {
"source_chapter": "Chapter 5 / 第五章 · 出版整理稿",
"source_reference": "Ch. 5 §§3.5–3.6",
"source_version": "QM-0accdcbe7725",
"source_sha256": "0accdcbe772504a352cb16ac5e41b6979599e8876b767666397faa8abccfb245",
"source_kind": "manifest",
"source_records": [
{
"id": "CH5-ca737ec4e327",
"title": "第五章-出版整理稿.docx",
"sha256": "ca737ec4e327d72c5507bbc8e00295b6e56029c7583ebad8e505cdb9c68c6262",
"format": "docx"
}
],
"source_manifest": "quantum-source-manifest.json",
"source_manifest_url": "/downloads/ASD_Quantum_Source_Manifest.json",
"sources": [
{
"label": "CH5-ca737ec4e327 · 第五章-出版整理稿.docx",
"kind": "supplied-manuscript"
}
]
}
}
---
## What width does a localized wave packet have?
A localized material core and its accompanying field do not have the same spatial distribution. The object here is the effective envelope: which wave-number components are required to form a field concentrated in a finite region? Reducing its spatial extent changes its phase gradients and spectrum. An uncertainty relation first constrains these two widths quantitatively; their connection to material measurements comes next.
Take a one-dimensional envelope $\psi(x)$, where $x$ is position along the direction considered. Normalize it and use the symmetric Fourier convention of Chapter 5, (5.3.23):
Here $k$ is the wave number conjugate to $x$, and $\widetilde\psi(k)$ is its amplitude. Their squared moduli have equal integral normalization and define mathematical weights in position and wave number. This does not yet identify them with experimental probabilities for core positions or momentum readings.
To give “width” a definite meaning, define standard deviations about the means:
$$
x_0=\int x|\psi(x)|^2dx,\qquad k_0=\int k|\widetilde\psi(k)|^2dk,
$$
$$
(\Delta x)^2=\int(x-x_0)^2|\psi(x)|^2dx,\qquad
(\Delta k)^2=\int(k-k_0)^2|\widetilde\psi(k)|^2dk.
$$
The second position moment must be finite, the first weak derivative of $\psi$ must be square-integrable, and the decaying boundary conditions used below must hold. A weak derivative allows envelopes that need not be classically differentiable everywhere but still have a suitable derivative in the integral sense. A specified slit width, half-maximum width or first-zero separation is not automatically this standard deviation.
## How integration by parts produces a lower bound
To bring spatial and spectral width into one calculation, define $F=(x-x_0)\psi$ and $G=(\partial_x-ik_0)\psi$, with inner product $\langle F,G\rangle=\int F^*G\,dx$. The function $F$ weights displacement from the center. The function $G$ removes the mean wave number's common linear phase from the derivative.
Taking the real part of the inner product removes the contribution from $-ik_0$ and leaves the derivative of the squared envelope modulus. The decaying boundary conditions make $[(x-x_0)|\psi|^2]_{-\infty}^{\infty}$ vanish, so
$$
\begin{aligned}
2\operatorname{Re}\langle F,G\rangle
&=\int(x-x_0)\partial_x|\psi|^2dx\\
&=-\int|\psi|^2dx=-1.
\end{aligned}
$$
The inner product therefore cannot become arbitrarily small. Cauchy–Schwarz bounds its modulus by the product of the two function norms, while Parseval converts the position-domain derivative norm into the wave-number variance:
$$
\|F\|^2=(\Delta x)^2,\qquad
\|G\|^2=\int(k-k_0)^2|\widetilde\psi(k)|^2dk=(\Delta k)^2,
$$
$$
(\Delta x)^2(\Delta k)^2
\ge|\langle F,G\rangle|^2
\ge\bigl(\operatorname{Re}\langle F,G\rangle\bigr)^2
=\frac14.
$$
Taking nonnegative square roots gives $\Delta x\Delta k\ge1/2$. This follows from normalization, Fourier structure and the boundary conditions, without assuming stochastic detector response.
## From wave number to momentum requires material matching
Turning wave-number bandwidth into momentum width requires the phase–momentum matching $p=\hbar k$ discussed on the [matter-wave page](/en/theory/quantum/matter-wave). With the same constant $\hbar$ throughout this matching, $\Delta p=\hbar\Delta k$. The existing equation registry records the mathematical proof and this physical connection together:
Thus $\Delta x\Delta k\ge1/2$ is an envelope relation, while $\Delta x\Delta p\ge\hbar/2$ initially constrains the position and canonical-momentum distributions associated with that envelope. ASD must connect the measured core momentum distribution to the wave-number distribution through specified preparation, core–field coupling and detection protocols. Substituting $\hbar$ does not remove the conditions on mode-action matching or experimental mapping.
## Why a Gaussian width and a slit width cannot be interchanged
The lower bound is attainable. Chapter 5, (5.3.26), gives a Gaussian envelope with linear phase:
Here $\sigma_x>0$ is the position standard deviation, $x_0$ is the envelope center and $k_0$ is the mean wave number. Its wave-number standard deviation is $1/(2\sigma_x)$, giving a width product of exactly $1/2$. Translation, a common phase or a linear phase does not change that minimum. Additional nonlinear phase, hard truncation or a non-Gaussian amplitude generally increases the product.
Slit experiments often use a different quantity: the width of a diffraction main lobe. The far-field amplitude of an ideal hard rectangular aperture is a sinc function, with sinc² intensity. Its tails make the second wave-number moment diverge. The first-zero separation can describe the main lobe but cannot be inserted as the standard deviation in this inequality. Real edge smoothing, finite material response and detector bandwidth alter the distribution and must be included in the same variance calculation.
This distinction keeps the aperture formulas in [interference and diffraction](/en/theory/quantum/interference-diffraction) consistent with the width bound. Geometric slit width, diffraction first-zero width and root-mean-square width answer different measurement questions.
## Duration and spectrum of a finite process
Time here is the metrological parameter recording the evolution of a process. A finite-duration amplitude also requires more than one frequency, but its width relation must follow the actual signal definition. For example, a rectangular time window of duration $T$ and carrier frequency $\omega_c$ gives
$$
A_T(\omega)=\int_{-T/2}^{T/2}e^{i(\omega-\omega_c)t}\,dt
=T\operatorname{sinc}\!\left[\frac{(\omega-\omega_c)T}{2}\right].
$$
Here $\omega$ is angular frequency. The first-zero half-width of the rectangular-window main lobe is $\Delta\omega_0=2\pi/T$. With the adopted energy–frequency matching, $\Delta E_0=\hbar\Delta\omega_0$ gives the exact width identity $\Delta E_0T=2\pi\hbar=h$, as stated after final equation (5.3.27). This is a first-zero width, not a variance: the rectangular-window spectrum has a divergent second moment, and $T$ is not a time standard deviation.
For a square-integrable time amplitude with finite root-mean-square duration and angular-frequency standard deviation, the same Fourier proof gives $\Delta t\Delta\omega\ge1/2$. Another familiar process is $\theta(t)e^{-t/(2\tau)}e^{-i\omega_ct}$, where $\theta(t)$ is the step function and $\tau$ is the intensity lifetime. Its Lorentz-shaped power spectrum has angular-frequency full width at half maximum $1/\tau$. Lifetime, time window, half-maximum width and standard deviation retain their distinct definitions.
## How wave-packet structure enters measurement
ASD connects the bandwidth of a finite accompanying structure to the field's phase and gradient structure. Apparatus back-action describes how preparation or detection coupling changes that structure. The envelope's Fourier constraint, the dynamics of measurement back-action and individual-record statistics are related questions with different calculation objects.
This page establishes an envelope inequality with defined widths and its material connection conditional on momentum matching. It does not thereby specify microscopic core trajectories or a universal time operator. Discrete observables such as spin need their own generators and algebra. Phase–momentum matching remains associated with [OP-QM-002](/en/open-problems/quantum/phase-momentum-locking), statistical universality with [OP-QM-004](/en/open-problems/quantum/born-statistics), and the conditional detector calculation is developed in [detection and statistics](/en/theory/quantum/detection-statistics).
## Derivation sources
Position and wave-number standard deviations and Gaussian equality are in Chapter 5, §3.5, (5.3.23)–(5.3.26). Finite processes and spectra are in §3.6, (5.3.27). The explicit integral definitions and proof steps expand the same registered relations without changing their theory status.