Wave Packets and Their Construction

Attosecond science is wave-packet engineering by hand: shaping spectral amplitude and phase is how the 2023 Nobel experiments clocked electron dynamics at the 10⁻¹⁸ s scale, the bandwidth theorem dictating that a 100-as pulse needs tens of eV of spectral width. Ultrafast electron microscopes fight the same reciprocity in reverse — compressing packets in time broadens their energy spread, blurring spectra. And every atom interferometer begins by preparing atomic Gaussian packets whose two faces are set by laser cooling ($\sigma_p$) and trap release ($\sigma_x$).

Strike a tuning fork and the sound is a pure, endless tone — the acoustic cousin of a plane wave. Strike a piano's low string, pedal down, and many neighbouring frequencies sound at once; they beat against each other, and the sound arrives in a lump. That is the whole trick of localization: waves of slightly different wavelength agree in one place — crest on crest, all in phase — and disagree everywhere else, trough cancelling crest. Add a narrow band of wavenumbers and you get a wide lump; add a broad band and the region of agreement shrinks to a needle.

Quantum mechanics adds only one ingredient to this ancient wave lore: through de Broglie's $p=\hbar k$, wavenumber means momentum. So "which wavelengths are in the mix" is not a technicality of synthesis — it is the particle's momentum distribution. The recipe function $\phi(k)$ that a sound engineer would call the spectrum, quantum mechanics calls the momentum-space wavefunction, the same object introduced for general states in Momentum-Space Wavefunction; here it takes its most concrete form. One state, two complete descriptions: $\psi(x)$ answers "where," $\phi(k)$ answers "how fast," and Fourier analysis is the dictionary between them. The concept-level tour of this construction in Free Particle and Wave Packets promised that machinery; now we build it for real.

Take the delta-normalized plane waves of Free Particle and Plane Waves and superpose them with an amplitude $\phi(k)$ — the continuum version of expanding in an orthonormal basis:

where: $\phi(k)$ — the complex amplitude density for wavenumber $k$ (momentum $p=\hbar k$); the $1/\sqrt{2\pi}$ — the symmetric convention that will make the inverse transform and Plancherel's theorem come out clean. Equation (1) is a synthesis: it assembles the position-space shape from momentum ingredients. If $\phi$ is sharply peaked at $k_0$, the packet is nearly monochromatic and very wide; if $\phi$ is broad, the components dephase within a short distance and the packet is narrow. The recipe can be run backwards. Given the initial shape $\psi(x,0)$ — prepared by a slit, a trap that was just switched off, an emission process — the amplitude of each ingredient is extracted by projecting onto the corresponding plane wave:

where: the only change from (1) is the sign in the exponent — analysis instead of synthesis. Together (1) and (2) say $\psi$ and $\phi$ are a Fourier-transform pair: neither is more fundamental, and either determines the other completely. This is Plancherel's theorem in action, and its physical backbone is proved in the first derivation card below. One consistency check makes the whole interpretation hang together. If $|\psi(x)|^2$ integrates to one (total probability), what does $|\phi(k)|^2$ integrate to?

where: the equality of the two integrals — Parseval's identity — is not an accident but a theorem. It licenses the Born-rule reading of the second face: $|\phi(k)|^2\,\mathrm{d}k$ is the probability that a momentum measurement finds $p=\hbar k$ in $[\hbar k,\hbar(k+\mathrm{d}k)]$. Normalize once, in either representation, and both are normalized forever.

Fourier pairs obey a seesaw. Compress $\psi(x)$ by a factor $\lambda$ — replace $\psi(x)$ by $\psi(\lambda x)$ — and the transform dilates: $\phi(k)\to\phi(k/\lambda)$ (up to normalization). Whatever squeezes one face stretches the other. Made quantitative with root-mean-square widths, the seesaw becomes the bandwidth theorem:

where: $\sigma_x$ and $\sigma_k$ — the rms widths of $|\psi(x)|^2$ and $|\phi(k)|^2$; the second form follows by $p=\hbar k$. Read the logic direction carefully: this is a theorem about any Fourier pair — radar pulses and audio filters obey it identically — and quantum mechanics inherits it the moment it declares $\psi,\phi$ to be the two faces of one state. Position–momentum uncertainty is not an extra postulate bolted onto wave mechanics; it is wave mechanics. (The operator-theoretic version, with its commutator machinery and its generalization to arbitrary observable pairs, is the business of the Generalized Uncertainty Principle in the formalism chapter.)

One family turns inequality (4) into an equality — the family you will meet in coherent states, path integrals, and every laser lab: the Gaussian. Start from the momentum recipe — a Gaussian bump of $1/e$-half-width $\sim1/a$ centred on a carrier $k_0$:

where: $a$ — the length scale that will turn out to be exactly the packet's rms position width; $k_0$ — the mean wavenumber, giving the packet its mean momentum $\hbar k_0$. The prefactor normalizes $\int|\phi|^2\mathrm{d}k=1$. What position shape does this recipe synthesize? The derivation card carries out the integral honestly.

The worked Gaussian rewards a moment of respect. Its three parameters answer the three physical questions: where is the particle ($\langle x\rangle=0$ here; add a factor $e^{-ik x_1}$ to $\phi$ to centre it at $x_1$), how fast is it ($\langle p\rangle=\hbar k_0$), and how sharply is each defined ($\sigma_x=a$, $\sigma_p=\hbar/2a$). At $t=0$ it is the most classical state quantum mechanics permits — both faces as narrow as the seesaw allows simultaneously. What happens next — the centroid gliding at the group velocity, the width growing as the components dephase — is precisely the subject matter of Group Velocity versus Phase Velocity and Wave Packet Spreading and Dispersion; the packet built here is their shared initial condition.

The payoff of the two-face structure is that momentum questions are answered directly in the momentum face — no derivatives, no operators, just read the distribution $|\phi(k)|^2$ like any probability density:

where: the formulas are the ordinary mean and variance of the distribution $|\phi|^2$, scaled by $\hbar$. For the Gaussian (5): $\langle p\rangle=\hbar k_0$ and $\sigma_p=\hbar/2a$, by inspection of a normal distribution — compare the same computation in position space, which requires integrating $\psi^*(-i\hbar\,\partial_x)\psi$. The representation in which an operator is diagonal is the representation in which its statistics are effortless; that lesson, learned here concretely, becomes a central principle of the formalism chapter.

Example 1 — an electron beam's coherence budget (confidence builder). An electron microscope prepares Gaussian packets of rms width $\sigma_x=1.0$ nm riding a 100 eV beam. Find the momentum spread, the velocity spread, and the fractional momentum spread.

Step 1. Minimum-uncertainty momentum spread: $\sigma_p=\dfrac{\hbar}{2\sigma_x}=\dfrac{1.055\times10^{-34}}{2\times10^{-9}}=5.3\times10^{-26}\ \mathrm{kg\,m/s}$.

Step 2. Velocity spread: $\sigma_v=\sigma_p/m_e=(5.3\times10^{-26})/(9.109\times10^{-31})=\boxed{5.8\times10^{4}\ \mathrm{m/s}}$.

Step 3. Beam momentum at 100 eV: $p_0=\sqrt{2m_eE}=\sqrt{2(9.109\times10^{-31})(100)(1.602\times10^{-19})}=5.4\times10^{-24}\ \mathrm{kg\,m/s}$ (speed $v_0=p_0/m_e=5.9\times10^{6}$ m/s).

Step 4. Fractional spread: $\sigma_p/p_0=(5.3\times10^{-26})/(5.4\times10^{-24})=\boxed{0.98\%}$. The packet is a nearly monochromatic wave a thousand crests long — which is exactly why plane-wave scattering amplitudes describe beam experiments so well.

Example 2 — the top-hat recipe and its heavy tails (exam level). A monochromator passes all wavenumbers in a flat band $|k-k_0|<\Delta$ with equal amplitude, $\phi(k)=1/\sqrt{2\Delta}$ inside the band and zero outside, with $\Delta=5.0\times10^{8}\ \mathrm{m^{-1}}$. Find the packet shape, the location of its first zeros, and its rms width.

Step 1. Synthesize with (1). Shifting to $u=k-k_0$: $\psi(x,0)=\frac{e^{ik_0x}}{\sqrt{4\pi\Delta}}\int_{-\Delta}^{\Delta}e^{iux}\mathrm{d}u=e^{ik_0x}\sqrt{\frac{\Delta}{\pi}}\;\frac{\sin(\Delta x)}{\Delta x}$.

Step 2. The envelope is a sinc. Check normalization: $\int|\psi|^2\mathrm{d}x=\frac{\Delta}{\pi}\int\frac{\sin^2\Delta x}{(\Delta x)^2}\mathrm{d}x=\frac{\Delta}{\pi}\cdot\frac{\pi}{\Delta}=1$. ✓

Step 3. First zeros where $\Delta x=\pm\pi$: $x=\pm\pi/\Delta=\pm(3.1416)/(5.0\times10^{8})=\boxed{\pm6.3\ \mathrm{nm}}$ — the central lobe is about 13 nm wide.

Step 4. Now the trap: the rms width. $\sigma_x^2\propto\int x^2\,\frac{\sin^2\Delta x}{(\Delta x)^2}\mathrm{d}x=\frac{1}{\Delta^2}\int\sin^2(\Delta x)\,\mathrm{d}x=\infty$: the rms width diverges. The sinc's tails fall only as $1/x^2$ in probability — too slowly for a second moment. The honest width statements are the central-lobe width $2\pi/\Delta$ or the half-power width, and the bandwidth theorem is respected trivially ($\infty>1/2$). Sharp edges in one Fourier face always buy slow decay in the other: the top-hat's rms $\sigma_k=\Delta/\sqrt3$ is perfectly finite while $\sigma_x$ is not.

Step 5. Moral: brick-wall spectral edges produce long oscillatory tails (diffraction ringing, the Gibbs phenomenon), which is why real monochromators taper their edges — and why the edge-free Gaussian is the shape with no ringing at all.

References. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §2.4 (the free particle) · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 5 (Gaussian wave packets) · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.4 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Complement G_I · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 6 Key relations underlying this article: the Fourier-pair normalization rests on the completeness relation $\frac{1}{2\pi}\int e^{i(k'-k)x}\mathrm{d}x=\delta(k'-k)$; both Gaussian derivations use the standard integral $\int e^{-\alpha k^2+\beta k}\mathrm{d}k=\sqrt{\pi/\alpha}\,e^{\beta^2/4\alpha}$; the Gaussian minimum-uncertainty product is $\Delta x\,\Delta k=1/2$ with $\Delta x=a$ and $\Delta k=1/2a$; and the time-dependent spreading width is $\sigma(t)=a\sqrt{1+(\hbar t/2ma^2)^2}$, with characteristic spreading time $\tau=2ma^2/\hbar$ — the latter is treated in full in Wave Packet Spreading and Dispersion.

Builds on: Free Particle and Plane Waves · Time-Independent Schrödinger Equation in One Dimension · Bound States versus Scattering States

Leads to: Group Velocity versus Phase Velocity · Wave Packet Spreading and Dispersion