Wave-packet spreading and uncertainty are two sides of the same Fourier logic. The packet moves with the group velocity, spreads because different \(k\) components evolve differently, and obeys a minimum product of position and momentum uncertainties.
The panels show the central Fourier tradeoff. When the position distribution \(|\Psi(x)|^2\) becomes wider, the wave-number distribution \(|\phi(k)|^2\) becomes narrower, and vice versa.
The initial packet from Item 2.13 evolves by giving each plane-wave component its free-particle phase:
This form is used because each component \(e^{ikx}\) has energy \(E=\hbar\omega(k)=\hbar^2k^2/2m\). Time evolution changes the relative phases of the components, and those changing phases reshape the packet in \(x\)-space.
These quantities are obtained from the time-dependent packet in the previous card. The phase \(kx-\omega(k)t\) inside \(\Psi(x,t)\) gives the phase velocity \(v_p\), also written \(v_{\mathrm{ph}}\), while the \(k\)-dependence of \(\omega(k)\) gives the group velocity \(v_g\):
The packet center moves with \(v_g\). At the central wave number \(k_0\), this gives \(v_g=\hbar k_0/m=p_0/m\), the classical velocity associated with the mean momentum.
Different \(k\) components have different \(\omega(k)\), so their phases do not remain locked together. The result is packet spreading. For the Gaussian packet,
The wave-number distribution \(|\phi(k)|^2\) does not change shape for a free particle, so
Position spreads with time, but the momentum spread stays fixed.
Combining the time-dependent position variance with the fixed momentum variance gives
At \(t=0\), the product is exactly \(\hbar/2\). For later times it is larger, consistent with the general Heisenberg relation:
The initial Gaussian saturates the bound; free evolution preserves the momentum width but increases the position width.
This page is designed to support short guided exercises on: Group velocity, phase velocity, time-dependent variance and the Heisenberg uncertainty principle.
- 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: Group velocity, phase velocity, time-dependent variance and the Heisenberg uncertainty principle.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\phi(k)\,e^{i[kx-\omega(k)t]}\,dk,\qquad \omega(k)=\frac{\hbar k^2}{2m}.\]
- Equation: \[v_p\equiv v_{\mathrm{ph}}=\frac{\omega}{k}=\frac{\hbar k}{2m},\qquad v_g=\frac{d\omega}{dk}=\frac{\hbar k}{m}.\]
- Equation: \[\mathrm{var}(x,t)=\frac{a^2}{2}\left[1+\left(\frac{t}{\tau}\right)^2\right],\qquad \tau=\frac{ma^2}{\hbar}.\]
- 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.