A plane wave is mathematically simple but physically delocalized. A localized particle-like packet requires a superposition of plane waves, and the distribution \(\phi(k)\) controls how sharp or spread the momentum information is.
For a free particle, \(V=0\), a plane wave has the form
It has a definite momentum, because it is an eigenfunction of \(\hat p\).
For a free particle, \(E=p^2/2m\). Using \(E=\hbar\omega\) and \(p=\hbar k\),
The frequency depends on \(k\), so different wave-number components in a packet evolve with different phases.
A plane wave extends through all space, so it does not represent a localized particle. It gives a sharp momentum but a completely spread-out position distribution.
The function \(\phi(k)\) is the wave-number distribution. A narrow \(\phi(k)\) gives a nearly definite momentum; a broad \(\phi(k)\) is needed to localize the packet in space.
A single plane wave has a definite wave number and therefore a definite momentum, but it extends through all space. That makes it a poor model for a localized particle.
A localized packet is built by superposing many nearby wave numbers. The narrower the packet in position, the broader the range of wave numbers required.
This page should support exercises about the tradeoff between localization and momentum spread, not full Fourier-analysis derivations.
This page is designed to support short guided exercises on: Why a single plane wave has sharp momentum but poor localization, and why packets are needed.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
- Keep in the book: complete derivations, extended historical discussion, worked solutions and the full textbook narrative remain in the original chapter and linked book resources.
- Good exercise balance: ask the student to identify assumptions, apply one relation, and interpret the result physically, without requiring material not introduced on this page.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.
- Focus: Why a single plane wave has sharp momentum but poor localization, and why packets are needed.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\Psi(x,t)=A e^{i(k_0x-\omega t)},\qquad p_0=\hbar k_0.\]
- Equation: \[\omega(k)=\frac{\hbar k^2}{2m}.\]
- Equation: \[\Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\phi(k)e^{i[kx-\omega(k)t]}\,dk.\]
- Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
- Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.