A free Gaussian wave packet whose position standard deviation starts at spreads…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch The path integral

Problem

A free Gaussian wave packet whose position standard deviation starts at σ0\sigma_0 spreads appreciably on the time scale τ=mσ02/ℏ\tau = m\sigma_0^2/\hbar, taken here up to a factor of order one. For an electron (m=9.11×10−31 kgm=9.11\times10^{-31}\,\mathrm{kg}) with σ0=1.0×10−10 m\sigma_0=1.0\times10^{-10}\,\mathrm{m} (an atomic-scale width): ((a)) Compute τ\tau. ((b)) Using the stationary-phase estimate for the width of the surviving bundle of paths after a time τ\tau, w=ℏτ/mw=\sqrt{\hbar\tau/m}, compute ww. Show that w=σ0w=\sigma_0 exactly, and explain why no other result was possible for a free particle.

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