Position–Momentum and Energy–Time Uncertainty in Quantum Systems

All flashcards

What does the de Broglie relation $p = \hbar k$ connect?

A particle’s momentum $p$ and its matter wave’s wave number $k$.

What is a wave packet?

A superposition of plane waves with different wavelengths and momenta that can localize a particle.

State the position–momentum uncertainty principle.

Position and momentum uncertainties satisfy $\Delta x\,\Delta p \ge \hbar/2$.

State the energy–time uncertainty relation.

Energy and lifetime uncertainties satisfy $\Delta E\,\Delta t \ge \hbar/2$.

How does an excited state’s lifetime affect its emission line?

Shorter-lived states have greater energy uncertainty and broader spectral lines.

Quiz questions

1. For an object of mass $m$ with speed uncertainty $\Delta u$, what minimum position uncertainty follows when $\Delta p = m\Delta u$?Answer: $\Delta x_{\min} = \hbar/(2m\Delta u)$
2. In the hydrogen energy estimate, why is the electron’s average momentum taken to be approximately zero?Answer: It is treated as equally likely to move in either direction.
3. For photons, how are energy spread $\Delta E$ and frequency spread $\Delta f$ related?Answer: $\Delta E = h\Delta f$
4. Which relation expresses the energy–time uncertainty bound?Answer: $\Delta E\,\Delta t \ge \hbar/2$
5. What is special about a Gaussian wave packet’s position–momentum uncertainty product?Answer: It can reach the minimum value, $\hbar/2$.