Playground
Hyperbolic boundary Δx·Δp = ℏ/2 with forbidden/allowed regions. Drag Δx to see Δp change.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Position uncertainty Standard deviation of position | m | L | 1e-18 – 0.001 | |
| Momentum uncertaintyoutput Standard deviation of momentum | kg*m/s | M*L*T^-1 | 1e-30 – 1e-18 | |
| Reduced Planck constant h / (2*pi) | J*s | M*L^2*T^-1 | 1.054571817e-34 – 1.054571817e-34 |
Deep dive
Derivation
Start from [x, p] = i*hbar. For any state |psi>, the Robertson inequality gives Delta_A * Delta_B ≥ |<[A,B]>|/2. Substituting [x,p] = i*hbar yields Delta_x * Delta_p ≥ hbar/2. A Gaussian wavefunction saturates the bound.
Experimental verification
Single-slit diffraction patterns, electron microscopy resolution limits, and precision measurements of squeezed light (LIGO) all confirm the relation. Ozawa-type uncertainty refinements measured directly with neutron spin experiments.
Common misconceptions
- The uncertainty is NOT due to measurement disturbance — it is an intrinsic property of the state
- It does NOT mean 'observer affects reality' in a mystical sense
- The factor of 2 matters: the lower bound is hbar/2, not h or hbar
Real-world applications
- Zero-point motion of atoms in crystals
- Squeezed-light interferometry (LIGO gravitational wave detection)
- Natural linewidth of atomic transitions (energy-time form)
- Fundamental limit on electron microscope resolution
Worked examples
Electron in a 1 Angstrom box
Given:
- Delta_x:
- 1e-10
Find: Delta_p_min, Delta_v_min
Solution
Delta_p ≥ 5.27e-25 kg*m/s; Delta_v ≥ 5.79e5 m/s
Energy-time: atomic level lifetime
Given:
- Delta_t:
- 1e-8
Find: Delta_E_min
Solution
Delta_E ≥ 5.27e-27 J ≈ 3.3e-8 eV (natural linewidth)
Scenarios
What if…
- scenario:
- What if hbar → 0?
- answer:
- Classical limit — position and momentum can be known simultaneously with arbitrary precision.
- scenario:
- What if we measure Delta_x to 10 fm (nuclear scale)?
- answer:
- Delta_p ≥ 5.27e-21 kg*m/s; Delta_v for an electron exceeds c, signalling relativistic QM is needed.
- scenario:
- What if we use a non-minimum uncertainty state?
- answer:
- Product Delta_x*Delta_p > hbar/2 strictly — e.g., higher excited states of the harmonic oscillator.
Limiting cases
- condition:
- Delta_x → 0
- result:
- Delta_p → ∞
- explanation:
- Perfectly localized particle has completely undetermined momentum.
- condition:
- Delta_x → ∞
- result:
- Delta_p → 0
- explanation:
- Plane-wave momentum eigenstate is completely delocalized in space.
- condition:
- Gaussian wave packet
- result:
- Delta_x*Delta_p = hbar/2 (equality)
- explanation:
- The minimum-uncertainty state — saturates the bound.
Context
Werner Heisenberg · 1927
Heisenberg derived this limit from matrix mechanics; Kennard and Robertson later formalized it as a variance inequality.
Hook
Can you know exactly where an electron is AND how fast it's moving?
Quantum mechanics says no — the product of position and momentum uncertainties has a hard lower bound. How precisely can you locate an electron confined to 1 Å?
Dimensions:
- lhs:
- [L] * [M*L*T^-1] → [M*L^2*T^-1]
- rhs:
- hbar → [M*L^2*T^-1]
- check:
- Both sides are action. ✓
Validity: Holds for any quantum state and any pair of non-commuting observables with [A,B] = i*hbar. Generalized form: Delta_A * Delta_B ≥ |<[A,B]>|/2.