K(x,T;0,0) — propagator from x₀=0
Feynman path integral: K(x,T;x₀,0) = ∫Dx(t) exp(iS[x(t)]/ℏ), summing over all paths.
Each path carries phase e^{iS/ℏ}; nearby paths interfere. The classical path (stationary action δS=0) dominates as ℏ→0 via stationary phase.
Free particle: K = √(m/2πiℏT) · exp(im(x−x₀)²/2ℏT). Harmonic oscillator: K ∝ exp(iS_cl/ℏ).
The spreading of |K|² = probability density is the quantum diffusion packet — uncertainty principle in action.
Each path carries phase e^{iS/ℏ}; nearby paths interfere. The classical path (stationary action δS=0) dominates as ℏ→0 via stationary phase.
Free particle: K = √(m/2πiℏT) · exp(im(x−x₀)²/2ℏT). Harmonic oscillator: K ∝ exp(iS_cl/ℏ).
The spreading of |K|² = probability density is the quantum diffusion packet — uncertainty principle in action.