AcademyPhotons
Academy
Uncertainty Principle
Level 1 - Physics topic page in Photons.
Principle
The uncertainty principle is a limit on quantum states, not a defect in measuring tools. A state cannot have arbitrarily precise position and momentum at the same time.
Localising a particle more tightly requires a larger spread in momentum.
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 or lifetime scale
\(\mathrm{s}\)
\(\hbar\)
reduced Planck constant
\(\mathrm{J\,s}\)
\(h\)
Planck constant
\(\mathrm{J\,s}\)
Method
Derivation 1: Position and momentum
Wave packets need a spread of wavelengths to be localized. A spread of wavelengths means a spread of momenta.
Uncertainty relation
\[\Delta x\,\Delta p\ge\frac{\hbar}{2}\]
Minimum momentum spread
\[\Delta p\ge\frac{\hbar}{2\Delta x}\]
Derivation 2: Energy and time
Short-lived states have an unavoidable spread in energy.
Energy-time relation
\[\Delta E\,\Delta t\gtrsim\frac{\hbar}{2}\]
Energy spread estimate
\[\Delta E\gtrsim\frac{\hbar}{2\Delta t}\]
Derivation 3: Connect to diffraction
A narrow slit localizes transverse position, but it produces transverse momentum spread and angular diffraction.
Aperture localization
\[\Delta y\sim a\]
Transverse momentum spread
\[\Delta p_y\sim\frac{h}{a}\]
Rules
Position momentum uncertainty
\[\Delta x\,\Delta p\ge\frac{\hbar}{2}\]
Momentum uncertainty
\[\Delta p\ge\frac{\hbar}{2\Delta x}\]
Energy time uncertainty
\[\Delta E\,\Delta t\gtrsim\frac{\hbar}{2}\]
Reduced Planck constant
\[\hbar=\frac{h}{2\pi}\]
Examples
Question
An electron is localized to
\[0.100\,\mathrm{nm}\]
Estimate the minimum momentum uncertainty using \[\hbar=1.05\times10^{-34}\,\mathrm{J\,s}\]
Answer
\[\Delta p\ge\frac{1.05\times10^{-34}}{2(1.00\times10^{-10})}=5.25\times10^{-25}\,\mathrm{kg\,m\,s^{-1}}\]
Checks
- Reducing position uncertainty increases the minimum possible momentum uncertainty.
- The uncertainty principle applies even with ideal instruments.
- The lower bound is a product, so either uncertainty can be large.
- Energy-time uncertainty is often used as a lifetime-linewidth estimate.