QM · Ch 3 · Square-well uncertainty product σ_x σ_p

The uncertainty principle isn't just a limit—it's a measurable consequence of how wave functions are shaped. Here, you'll calculate how "spread out" the particle's position and momentum are around their average values, then see that the ground state's compact, smooth shape naturally produces an uncertainty product right at the quantum limit. The key is recognizing that $\sigma_x$ and $\sigma_p$ both follow from integrals of the probability density $|\psi|^2$, and the ground state's single hump minimizes their product.

The uncertainty principle isn't just a limit—it's a measurable consequence of how wave functions are shaped. Here, you'll calculate how "spread out" the particle's position and momentum are around their average values, then see that the ground state's compact, smooth shape naturally produces an uncertainty product right at the quantum limit. The key is recognizing that $\sigma_x$ and $\sigma_p$ both follow from integrals of the probability density $|\psi|^2$, and the ground state's single hump minimizes their product.