Heisenberg Wavepacket & Uncertainty Simulator

Spatial Width (Δx) 1.500 a.u. Position uncertainty

Momentum Spread (Δp) 0.333 a.u. Fourier conjugate uncertainty

Uncertainty Product 0.500 ħ ✓ Minimum Uncertainty State

Fourier Components 64 waves Superposition count

Real-Space Wavefunction Ψ(x, t)

Drag center x₀ or Δx slider

Cyan: Re(Ψ) | Magenta: Im(Ψ) | Amber Area: |Ψ|²

Momentum-Space Spectrum Φ(p)

Fourier Transform Distribution

Amber Curve: |Φ(p)|² Momentum Probability

Wavepacket Parameters Real-time Fourier Synthesis

Spatial Uncertainty (Δx) 1.50

Central Momentum (p₀) 2.00

Center Position (x₀) 0.00

Potential Barrier Height (V₀) 0.00

Physical Scenarios

📖 Quantum Philosophy & Mathematical Foundation

In Physics and Philosophy (1958), Werner Heisenberg explained that quantum mechanics fundamentally changed our view of objective reality. The Uncertainty Principle is not merely a limitation of experimental apparatus—it is an intrinsic property of wave-particle duality.

Δx · Δp ≥ ħ / 2 | Ψ(x) = (1 / (2π Δx²)^(1/4)) · exp(-(x - x₀)² / (4 Δx²)) · exp(i p₀ x / ħ)

When you compress the spatial footprint Δx of a wavepacket, its constituent momentum Fourier spectrum Φ(p) spreads out proportionally. Clicking Simulate Measurement collapses the probability distribution |Ψ(x)|² into a single localized position eigenstate, demonstrating how observation alters the physical system.

Reading the wavepacket parameter formulas

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Read the explanation

This page assigns the momentum spread from the chosen position spread. Its units are arbitrary, and h bar is set to one. Begin at position spread point three, giving momentum spread one point six seven. Increase the position parameter to four. The source formula divides one by twice that input, so the momentum parameter falls to point one two five. The moving point follows that reciprocal relation. Their product stays point five. These are assigned Gaussian parameters, not variances numerically measured from the animated canvas. Now examine only the Gaussian envelope used by the real space drawing. Set initial width one point five and central momentum zero. At the initial internal time, the width is one point five. Compare internal time zero through six. The source multiplies the initial width by the square root of one plus time over twice the squared initial width, all squared. This increases the width to two point five. The animation adds point zero one five per rendered frame. This horizontal axis is an internal parameter comparison, not seconds on a physical clock, and this envelope alone does not establish a complete wave equation solver. Keep the internal time at three, mass at one, and initial center zero. The drawing computes its Gaussian center as initial center plus central momentum times internal time. At momentum minus two, the center is minus six. Raise central momentum from minus two to plus two. The center moves along this straight relation and reaches plus six. We compare settings at fixed internal time, rather than claiming a measured trajectory. This lesson covers the assigned reciprocal spread, envelope width, and center formulas. It does not validate the barrier or measurement branches, or infer physical scattering from their display.