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}

where

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

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