Uncertainty Principle

Valid width and initial phase show the initial packet immediately. Edits to σ or c stop playback and return to t=0. Type a model time to inspect it directly. Play/Pause advances time explicitly; Reset returns to t=0 with the same packet. For c<0, Focus selects its minimum position spread. Options contains playback rate, the initial-position comparison and a manual +0.1 step. Blank, invalid, comma and leading-zero drafts of σ,c and playback rate are retained, together with the comparison choice. Legacy numeric settings migrate. Time, playback, curves and results are not stored; reloading starts at t=0. Hidden tabs and reduced motion stop animation while manual time and Focus remain available. The interval is 0≤t≤20; playback rate is model time per screen second, not a physical clock.

Model: centered normalized pure Gaussian, free Hamiltonian H=p²/2, ħ=m=1, zero mean position and momentum. Initially ψ(x,0)=(2πσ²)⁻¹/⁴ exp[−x²/(4σ²)+icx²/(4σ²)]. Thus σ is the standard deviation of |ψ|², not of the amplitude. Plots show normalized probability densities, not amplitudes; their areas over the full real line equal1. Model units have no chosen SI mass/length scale. Dashed profiles are initial distributions; momentum density remains unchanged under free evolution. Numerical axes auto-fit the displayed curves; pixel widths alone are not a common unit scale.

Δp=√(1+c²)/(2σ), Cₓₚ(0)=c/2. At time t, Δx²=σ²+ct+t²Δp² and Cₓₚ=c/2+tΔp². Consequently ΔxΔp≥1/2 and Δx²Δp²−Cₓₚ²=1/4. For c=0,t=0 the usual minimum product is attained at any width. Free evolution generally increases position–momentum correlation, so the simple product can grow although the state remains a pure Gaussian saturating the stronger covariance bound. No measurement disturbance is being simulated.

Negative c gives initial contraction. Focus chooses t*=−c/(2Δp²), where covariance vanishes and the product returns to1/2; the packet then expands. Positive c initially expands. The initial chirp c changes momentum spread without changing initial position density. No potentials, scattering, finite FFT grid, mixed states or experimental noise are modelled. This is an analytic family of solutions, not a numerical propagation benchmark.

MIT: Time Evolution · Robinett, Doncheski, Bassett: Correlated Gaussian wavepackets