State the position-momentum uncertainty relation.

What does \(\Delta x\) represent in the uncertainty relation?

What happens to minimum momentum uncertainty when a particle is localized more tightly?

An electron is localized to \(1.00\times10^{-10}\,\mathrm m\). Estimate the minimum \(\Delta p\).

If \(\Delta x\) is halved, what happens to the minimum \(\Delta p\)?

State a common approximate energy-time uncertainty relation.

A state has lifetime \(1.0\,\mathrm{ns}\). Estimate its minimum energy width.

Convert \(5.3\times10^{-26}\,\mathrm J\) to electronvolts.

Why is uncertainty not just poor measurement technique?

Estimate the kinetic-energy scale for an electron confined to \(0.100\,\mathrm{nm}\) using \(\Delta p=5.25\times10^{-25}\,\mathrm{kg\,m\,s^{-1}}\).

Derive \(\Delta p\ge\hbar/(2\Delta x)\) from the uncertainty relation.

Why does confinement raise the ground-state energy of a particle?

A particle is confined to a region twice as wide. How does the uncertainty-based confinement energy scale change?

Compare uncertainty effects for an electron and proton confined to the same length.

Why does a shorter excited-state lifetime imply a broader spectral line?

Derive the confinement energy scale \(K\sim\hbar^2/(2mL^2)\).

Explain why an electron cannot be exactly at rest inside a finite atom.

How does uncertainty help explain why atoms have finite size?

Why is energy-time uncertainty interpreted differently from position-momentum uncertainty?

Connect de Broglie waves and uncertainty in one explanation.