QNT-05

Heisenberg uncertainty

Δx Δp ≥ ħ/2; given Δx, the bound on Δp.

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UncertaintyHeisenberg 1927

Governing equation

\Delta x\,\Delta p\ge\dfrac{\hbar}{2}

$\Delta x \in \left[0.001, 10\right]$

where

\Delta x: Position σ (nm)
\Delta p_{\min}: Min momentum σ (keV/c)
\Delta v_{\min}: Electron Δv min (km/s)

The figure, part by part

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①

Governing law — red

\Delta x\,\Delta p\ge\dfrac{\hbar}{2}

$\Delta x \in \left[0.001, 10\right]$

$\Delta x = 0.001$ $\Delta p_{\min} = 98.66$

Along the curve

②

Domain of the variable — blue

\Delta x_{\min} = 0.001,\quad \Delta x_{\max} = 10

$\Delta x \in \left[0.001, 10\right]$

$\Delta x = 0.001$ $\Delta x = 0.001$

Along the curve

③

Response curve — green

\Delta x\,\Delta p\ge\dfrac{\hbar}{2}

$\Delta x \in \left[0.001, 10\right]$

$\Delta x = 0.001$ $\Delta p_{\min} = 98.66$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta p_{\min}}{\mathrm{d}\Delta x}

$\Delta x \in \left[0.001, 10\right]$

$\Delta x = 0.064$ $\Delta p_{\min}^{\prime} = -778.3$

Along the curve

Lecture brief

Historical brief

Planck (1900), Einstein’s photoelectric law, Bohr, de Broglie, Heisenberg and Schrödinger’s 1926 equation rebuilt matter as amplitude. These sheets are the first solvable models: wells, spin, tunneling, uncertainty. This sheet (QNT-05 — Heisenberg uncertainty) is the form associated with Heisenberg 1927. Working symbols: $\Delta x$ $\rightarrow$ $\Delta p_{\min}$ , $\Delta v_{\min}$ . Non-commuting operators x and p imply a product of standard deviations bounded by ħ/2, the Robertson–Schrödinger form of Heisenberg's principle.

Purpose

Purpose: compute

\Delta p_{\min}

\Delta v_{\min}

from

\Delta x

in Quantum mechanics via

\Delta x\,\Delta p\ge\dfrac{\hbar}{2}

Δx Δp ≥ ħ/2; given Δx, the bound on Δp. Use it when a real quantum mechanics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\Delta x = 0.100\,\mathrm{nm}

, the governing relation

\Delta x\,\Delta p\ge\dfrac{\hbar}{2}

yields

\Delta p_{\min} = 0.9866\,\mathrm{keV/c}

\Delta v_{\min} = 578.838\,\mathrm{km/s}

. Minimum-uncertainty bound, 1-D. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Position σ \Delta x0.1 nm

Outputs

Min momentum σ \Delta p_{\min}0.9866 keV/c
Electron Δv min \Delta v_{\min}578.838 km/s

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Narration of this film

Minimum-uncertainty bound, 1-D.

Non-commuting operators x and p imply a product of standard deviations bounded by ħ/2, the Robertson–Schrödinger form of Heisenberg's principle.

Reading speed

Watch on YouTube