AcademyMatter Waves
Academy
Uncertainty Revisited
Level 1 - Physics topic page in Matter Waves.
Principle
Wave packets require spreads in position and momentum, producing the uncertainty relation.
Notation
\(\Delta x\)
position uncertainty
\(\mathrm{m}\)
\(\Delta p\)
momentum uncertainty
\(\mathrm{kg\,m\,s^{-1}}\)
\(\Delta E\)
energy uncertainty
\(\mathrm{J}\)
\(\Delta t\)
time interval
\(\mathrm{s}\)
\(\hbar\)
reduced Planck constant
\(\mathrm{J\,s}\)
\(m\)
particle mass
\(\mathrm{kg}\)
Method
Derivation 1: Connect localization and wave number spread
A sharply localized wave packet requires many wavelengths. That creates a spread in momentum.
Position momentum
\[\Delta x\Delta p\ge\frac{\hbar}{2}\]
Momentum estimate
\[\Delta p\gtrsim\frac{\hbar}{2\Delta x}\]
Derivation 2: Estimate confinement energy
If a particle is confined to width \(L\), its momentum uncertainty gives a minimum kinetic-energy scale.
Confinement momentum
\[\Delta p\sim\frac{\hbar}{L}\]
Energy scale
\[K\sim\frac{(\Delta p)^2}{2m}\]
Derivation 3: Use energy-time uncertainty carefully
Energy-time uncertainty estimates linewidths and lifetimes; it is not caused by a faulty clock.
Rules
Position momentum
\[\Delta x\Delta p\ge\frac{\hbar}{2}\]
Energy time
\[\Delta E\Delta t\gtrsim\frac{\hbar}{2}\]
Confinement scale
\[K\sim\frac{\hbar^2}{2mL^2}\]
Examples
Question
An electron is localized to
\[0.100\,\mathrm{nm}\]
Estimate the minimum \[\Delta p\]
Answer
\[\Delta p\ge\frac{\hbar}{2\Delta x}=\frac{1.05\times10^{-34}}{2(1.00\times10^{-10})}=5.25\times10^{-25}\,\mathrm{kg\,m\,s^{-1}}\]
Checks
- Uncertainty is not just instrument error.
- Smaller confinement length means larger momentum spread.
- Zero kinetic energy is incompatible with exact confinement.
- Energy-time uncertainty is used differently from position-momentum uncertainty.