10.2 Wave packets and uncertainty

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §10.2. Rewritten in original wording. Continues 10.1b. Next: 10.2b.

A finite sine burst

A wave packet is a pulse built from many frequencies so that it lives only for a while $\Delta t$. Its Fourier transform is then a pulse of width $\Delta\omega$ in frequency.

Prototype: $N$ cycles of $\sin\omega_0 t$, then zero:

$$y(t)=\begin{cases} \sin\omega_0 t, & |t|<N\pi/\omega_0,\\ 0, & \text{otherwise}. \end{cases}$$

Time width $\Delta t\approx N\cdot 2\pi/\omega_0$. The transform $Y(\omega)$ peaks at $\omega_0$; measuring width by the first zeros gives $\Delta\omega\approx\omega_0/N$.

Uncertainty bound

Combining $\Delta t\approx N\cdot 2\pi/\omega_0$ with $\Delta\omega\approx\omega_0/N$ forces a lower bound on the product — pick $\Delta t\,\Delta\omega\gtrsim 2\pi$ and the general $2\pi C$ form among lookalikes.

Sharper in time $\Rightarrow$ broader in frequency (and conversely). A pure eternal sine has $\Delta\omega\to 0$ and $\Delta t\to\infty$. Quantum mechanics writes the same story for conjugate observables; here it is a Fourier fact that also limits wavelet / STFT tiles. Estimate $C$ from FWHM measures on $|y|$ and $|Y|$ for Gaussians, Morlets, Mexican hats, etc.

Exercise menu (do in labs / notebook)

For $y_1=e^{-t^2/2}$, $y_2=\sin(8t)\,e^{-t^2/2}$, $y_3=(1-t^2)e^{-t^2/2}$: estimate $\Delta t$ (e.g. FWHM), DFT to get $Y(\omega)$ (linear + semilog), estimate $\Delta\omega$, then $C\approx\Delta t\Delta\omega/(2\pi)$.

Quiz

Wave packets

1 / 4

For an N-cycle sine burst at ω₀, a useful Δω estimate is…

Pick an answer to see if you are right.

Next: 10.2b Wave-packet labs.