Path Integral Lab

The real-time action is S=∫[½mẋ²−V(x)]dt, with m=1 and total time T=1 in dimensionless teaching units. The midpoint discretization uses28 equal time intervals. Each of27 internal positions is sampled independently and uniformly within the chosen spread around the straight bridge from A to B. The seed reproduces the complete sample. These are equal-weight finite-box samples, not a continuum path-integral calculation. All blue paths use the same opacity; none is artificially favored for being close to the classical action.

The complex vectors are exp(iS/ħ). Individual vectors have unit length; the orange vector is their arithmetic mean with its true length, between0 and1. Changing ħ rephases the same sample. Included paths selects a prefix; Reveal optionally animates this selection. Zero selected paths have an undefined mean. New seed increments the integer seed and generates a new realization. Histogram axes stay fixed while revealing. The mean is neither a normalized propagator nor a transmission probability; increasing N alone does not remove finite-box or discretization bias.

V is zero, harmonic ½k(x−½)² or a rectangular barrier of height V₀ and chosen width centered at½. The red continuous classical reference solves the free or harmonic boundary-value problem. At conjugate oscillator times the reference may be absent or nonunique, so no arbitrary curve is drawn. For the barrier, the reference is a monotone transmitted trajectory with energy conserved across interfaces, determined by its travel time; reflected branches are not enumerated. Equal endpoints away from a step admit the constant reference. The displayed classical action is the continuous action, while sampled actions use finite midpoint quadrature. Classical action is stationary, not necessarily a minimum. This demonstration does not measure tunnelling or prove classical-path dominance from a few hundred arbitrary paths.

Valid endpoint, potential and sampling changes update immediately, including the initial view. Click the left or right half of the path graph to change A or B, or enter them exactly (−2…2). Options contains seed, sample count/spread, potential parameters and extra plots. ħ and display changes retain the same paths; changing sampling parameters rebuilds them with the selected seed. Reset restores the live default. Only exact endpoint/seed input text and parameter/display choices are retained, including unfinished input. Generated paths, actions, sums and reveal position are not saved; reload reconstructs the full sample. Reference and sampled paths remain inside the automatic position range. Reveal is optional, pauses when hidden and respects reduced motion.

Tong, Classical Dynamics §4.8; Advanced Quantum Field Theory §§1.2–1.2.1.