Heisenberg Product σ_x σ_p
Gaussian wavefunctions minimize the Robertson uncertainty product for position and momentum: σ_x σ_p = ħ/2 for a minimum-uncertainty packet. With ħ set to 1, the simulator fixes σ_x and introduces a multiplicative “momentum excess” over the minimal σ_p = 1/(2σ_x). The product σ_x σ_p grows linearly with that factor. Two bar charts show |ψ(x)|² and |φ(p)|² with matched widths, reinforcing that narrowing one distribution widens the other unless excess is added.
Who it's for: Introductory quantum classes linking Fourier width intuition to the Heisenberg relation.
Key terms
- Heisenberg Uncertainty
- Gaussian Wavepacket
- Standard Deviation
- Momentum Space
How it works
Position and momentum widths for minimum-uncertainty Gaussians (ℏ = 1). Slide σ_x and an extra momentum spread to see the Heisenberg product σ_x σ_p stay at or above ℏ/2 with matched |ψ(x)|² and |φ(p)|² plots.
Frequently asked questions
- Does increasing “momentum excess” change the position width?
- No. σ_x is controlled independently to stress that σ_x σ_p can exceed ħ/2 when the state is no longer a single minimum-uncertainty Gaussian in both spaces simultaneously (here modeled as a wider momentum Gaussian paired with the same position Gaussian — a pedagogical illustration rather than a single Schrödinger eigenstate of free space).
More from Chemistry
Other simulators in this category — or see all 38.
β⁻ Decay: Electron Spectrum
Continuous kinetic-energy spectrum up to endpoint Q (allowed-decay phase space); missing energy carried by the antineutrino.
Nernst Equation
E = E° − (RT/nF) ln Q: sliders for E°, n, T, and reaction quotient.
Buffer Solution
Henderson–Hasselbalch vs strong acid: pH curve as H⁺ is added (mole model).
Gray–Scott Patterns
Reaction–diffusion u,v; coral / mitosis / worms / spirals; D_u, D_v, Δt.
Gibbs Free Energy
ΔG = ΔH − TΔS; sign vs spontaneity at constant p,T (no Q or K).
Maxwell–Boltzmann vs Eₐ
Translational energy density f(E); shaded fraction above activation energy; compare Arrhenius exp(−Eₐ/RT).