quantum mechanics
UNCERTAINTY PRINCIPLE
Interactive simulation of the Heisenberg Uncertainty Principle. Visualize the fundamental limit of simultaneous position and momentum knowledge.
Position Space ψ(x)
The wave packet in position space. |ψ(x)|² shows the probability density (red area). Re(ψ) shows the oscillating wave inside the envelope.
Momentum Space φ(k)
The Fourier transform into momentum space. As the position space narrows, momentum space broadens. Δx · Δk = 1/2
Uncertainty Relation
The uncertainty product Δx·Δp as a 2D rectangle. As one dimension shrinks, the other grows, keeping the area constant at ℏ/2.
Quantum Statistics
Position Uncertainty
1.00
Δx (σ)
Momentum Uncertainty
0.50
Δp (ℏ·Δk)
Wavevector Uncertainty
0.50
Δk (1/2σ)
Min. Uncertainty
0.50
ℏ/2
Uncertainty Product
0.50
Δx·Δp (ℏ/2 for Gaussian)
Time Domain ψ(t)
A wave packet with finite duration Δt. Shorter pulses have broader frequency spectra.
Frequency Domain φ(ω)
The frequency spectrum. ΔE = ℏ·Δω represents the energy spread. Shorter time → broader energy spectrum.
Energy-Time Uncertainty
Time Duration
1.00
Δt
Energy Spread
0.50
ΔE = ℏ·Δω
Energy-Time Product
0.50
ΔE·Δt (ℏ/2)
This relation explains why particles created in high-energy collisions are very short-lived: a short lifetime Δt implies a large energy spread ΔE. Conversely, stable particles have well-defined energies but indefinite lifetimes.
Single Slit Setup
Narrow slit = well-defined position (small Δx) but causes large diffraction (large Δp spreading).
Diffraction Pattern
The diffraction pattern shows how momentum uncertainty grows. Narrower slit → wider diffraction pattern. This IS the uncertainty principle in action!
Diffraction Analysis
Slit Width
1.00
Δx (a)
Angular Spread
0.50
θ₁st (radians)
Momentum Uncertainty
1.00
Δp (approx)
Uncertainty Product
0.50
Δx·Δp
Single-slit diffraction demonstrates the uncertainty principle directly. Confining a particle to a slit of width a (Δx ≈ a) forces a momentum uncertainty of roughly ℏ/a (Δp ≈ ℏ/a), giving Δx·Δp ≈ ℏ.