Wave Packets, Group Velocity and Spreading

Derivation

Wave Packets, Group Velocity and Spreading

D-072 Home PU-103 Threads waves Depends on Dispersion, Phase Velocity and Group Velocity, Fourier Decomposition into Normal Modes

Statement

A superposition of harmonic waves whose amplitude A(k) is peaked about a central wavenumber k₀ forms a localized packet. To first order in the spread of k the envelope translates rigidly at the group velocity v_g = (dω/dk)|_k₀ while the underlying carrier moves at the phase velocity v_p = ω₀/k₀; to second order the curvature β = (d²ω/dk²)|_k₀ makes the packet broaden, a Gaussian packet of initial r.m.s. width σ₀ reaching width σ(t) = σ₀√(1 + (βt/2σ₀²)²).

Why it matters

The packet is the bridge between a single idealized plane wave, which carries no information because it is infinite and monochromatic, and a physical signal, which is localized and therefore made of many frequencies. Group velocity is the speed at which energy, information and the probability density actually travel, and it is generically different from the phase velocity you read off a single crest.

Dispersive spreading sets hard limits everywhere: the reach of an optical fibre before pulses overlap, the coherence time of an electron wave packet in a microscope, the way an ocean swell sorts itself into long trains, and the delocalization of a free quantum particle. All of these follow from the same second-order term derived below.

Assumptions

A well-defined spectrum A(k) exists (the packet is Fourier-synthesizable).If the field cannot be written as an integral over modes — for a nonlinear medium where modes couple — superposition fails and the whole construction collapses.

A(k) is sharply peaked about k₀, i.e. its width Δk ≪ k₀.If the spectrum is broad, the Taylor expansion of ω(k) cannot be truncated and no single v_g or β describes the motion.

The medium is linear and time-independent, so each mode k evolves as e^−iω(k)t with a fixed dispersion relation ω(k).If ω depends on amplitude or the medium changes in time, the phases no longer add coherently and the envelope is not rigid.

Third- and higher-order dispersion (d³ω/dk³, …) is negligible over the propagation considered.If retained, the packet not only broadens but also skews and develops oscillatory tails; the clean Gaussian width law fails.

The group velocity is real, i.e. the medium is lossless (no imaginary part in ω(k)) near k₀.In an absorbing or gain region v_g can exceed c or go negative and stops representing energy transport.

Derivation

ψ(x,t) = (1/2π) ∫ A(k) e^{i(kx − ω(k)t)} dk

Fourier synthesis: any localized solution of a linear wave equation is a superposition of its normal modes, each evolving with its own ω(k). Uses fourier-series-mode-decomposition. A

ω(k) = ω₀ + v_g(k − k₀) + ½ β (k − k₀)² + …

Taylor-expand the dispersion relation about the spectral peak; legal because A(k) is sharp so only k near k₀ contribute. Definitions v_g = dω/dk|₀, β = d²ω/dk²|₀. Uses dispersion-phase-group-velocity. B

κ ≡ k − k₀, kx − ωt = (k₀x − ω₀t) + κ(x − v_gt) − ½ β κ² t

Substitute the expansion of step 2 into the exponent of step 1 and collect terms in powers of κ; pure algebra. B

ψ(x,t) = e^{i(k₀x − ω₀t)} u(x,t), u(x,t) = (1/2π) ∫ A(k₀+κ) e^{iκ(x − v_gt) − i½βκ²t} dκ

Factor the κ-independent carrier e^{i(k₀x−ω₀t)} out of the integral; it travels at v_p = ω₀/k₀. The remaining integral u is the envelope. A

β → 0: u(x,t) = (1/2π) ∫ A(k₀+κ) e^{iκ(x − v_gt)} dκ = u(x − v_gt, 0)

Drop the quadratic term; the envelope then depends on x and t only through x − v_gt, so it is rigidly translated at the group velocity — no change of shape. B

A(k₀+κ) = √(4πσ₀²) · e^{−σ₀²κ²}

Specialize to a Gaussian packet: this spectrum is the Fourier transform of the initial envelope u(x,0) = e^{−x²/4σ₀²}, for which |u|² has variance σ₀². Chosen because Gaussians stay Gaussian and give a closed form. B

u(x,t) ∝ ∫ e^{−aκ² + iκξ} dκ = √(π/a) · e^−ξ²/4a, ξ ≡ x − v_gt, a ≡ σ₀² + i βt/2

Complete the square in the Gaussian integral (valid for Re a > 0) combining the real spectral width σ₀² with the imaginary dispersive phase iβt/2. C

|u|² ∝ e^{−ξ² Re(1/2a)} = e^{−ξ²/2σ(t)²}, σ(t)² = (σ₀⁴ + β²t²/4)/σ₀²

Take the modulus squared, using Re(1/2a) = σ₀²/[2(σ₀⁴+β²t²/4)], and read off the time-dependent variance. C

Result

envelope centre: x = v_gt • σ(t) = σ₀ √(1 + (βt / 2σ₀²)²) • v_g = dω/dk, β = d²ω/dk²

Reading. The packet's centre of mass moves at the group velocity v_g, not the phase velocity of individual crests. Its width stays essentially constant for times short compared with the spreading time τ = 2σ₀²/|β|, then grows linearly, σ(t) ≈ |β|t/2σ₀, at long times. A narrow packet (small σ₀, hence broad spectrum) spreads fastest — the price of tight localization.

Units check. β = d²ω/dk² has units (rad s⁻¹)/(rad m⁻¹)² = m² s⁻¹. Then βt/2σ₀² is (m² s⁻¹ · s)/m² = dimensionless, so the square root is dimensionless and σ(t) carries the units of σ₀, i.e. metres. The centre position v_gt is (m s⁻¹)(s) = m. Consistent.

Limiting cases

Non-dispersive medium, ω = ck: v_g = v_p = c and β = 0, so σ(t) = σ₀ forever — the packet moves at c without changing shape (e.g. light in vacuum).
Short times t ≪ τ: σ(t) ≈ σ₀[1 + ½(βt/2σ₀²)²], broadening is quadratic and negligible; the rigid-translation picture of step 5 holds.
Long times t ≫ τ: σ(t) → |β|t/2σ₀, ballistic spreading — width grows linearly in t with rate set by the spectral width 1/σ₀.
Free quantum particle, ω = ℏk²/2m: v_g = ℏk₀/m = p/m (the classical velocity) and β = ℏ/m, giving σ(t) = σ₀√(1 + (ℏt/2mσ₀²)²).

Breaks when

Broad spectrum (Δk not small). When the packet is only a few wavelengths wide the Taylor truncation is invalid; higher derivatives of ω(k) matter and neither a single v_g nor the Gaussian width law applies.
Strong or resonant dispersion / absorption. Near an absorption line ω(k) is complex and steeply varying; v_g can exceed c or reverse sign, and the second-order model gives meaningless (superluminal energy transport) predictions.
Nonlinear media. If the refractive index depends on intensity (Kerr effect), modes couple; effects such as self-phase modulation and solitons can cancel or reverse the linear spreading, so σ(t) above no longer holds.

Failure modes

Confusing v_p and v_g: quoting ω/k as the signal speed. The crests move at v_p; the packet (and its energy) moves at v_g = dω/dk.
Sign errors in β: worrying that a negative β means the packet narrows. Only β² enters σ(t), so anomalous and normal dispersion both broaden a chirp-free packet.
Wrong dispersion quantity: using the optics parameter β₂ = d²k/dω² (units s² m⁻¹) directly in the width formula, which requires β = d²ω/dk² (units m² s⁻¹). They are not equal.
Forgetting the initial-width dependence: thinking a narrower packet is always "better." Narrower σ₀ means faster spreading — there is an optimal σ₀ minimizing width at a target distance.
Truncating too early: dropping β and then being surprised the pulse smears — the linear term alone can never spread the packet.

Discussion

The clean split into a carrier e^{i(k₀x−ω₀t)} times a slow envelope u(x,t) is the origin of the "slowly varying envelope" method used throughout optics and plasma physics. It works precisely because the spectrum is narrow: the carrier oscillates on the scale 1/k₀ while the envelope varies on the much larger scale 1/Δk = σ₀. The group velocity emerges as the velocity of the first non-trivial feature of the envelope, and it is a stationary-phase result: the packet is largest where the phases of neighbouring modes agree, d(kx−ωt)/dk = 0, giving x = (dω/dk)t.

Spreading is a direct consequence of the uncertainty-like relation Δx · Δk ≳ 1. Because the packet contains a range of wavenumbers Δk ∼ 1/σ₀, and each travels at a slightly different group velocity v_g(k) ≈ v_g + β(k−k₀), the spread in speeds is Δv ∼ β/σ₀. Over time t the fastest and slowest components separate by Δv · t ∼ βt/σ₀, exactly the long-time limit of σ(t). Dispersion is nothing more than different frequencies keeping different appointments.

The quantum case is especially instructive. The free-particle result σ(t) = σ₀√(1 + (ℏt/2mσ₀²)²) shows that a localized particle inevitably delocalizes, and that heavier particles spread more slowly (β = ℏ/m). This is why a dust grain stays put while an electron smears over nanometres in femtoseconds, and it is the same mathematics that governs a chirped laser pulse — the free Schrödinger equation and the paraxial/dispersive wave equation are formally identical, both diffusion equations with imaginary diffusion constant.

Deeper still, the imaginary "diffusion constant" iβ/2 in step 7 makes the spreading unitary rather than dissipative: no probability (or energy) is lost, it is merely redistributed, and the process is time-reversible. A packet that has spread will re-focus if the sign of β is reversed and the phase relationships preserved — the principle behind dispersion-compensating fibre, chirped-pulse amplification and spin/photon echoes. This is qualitatively different from genuine diffusion, where an increasing-entropy real diffusion constant makes the process irreversible.

Common misconceptions. A packet does not spread because "waves lose energy" or "the medium absorbs high frequencies" — lossless dispersion conserves the total ∫|ψ|²dx exactly. It spreads purely because its component waves travel at different speeds. Nor does the carrier "carry the signal": switch off the dispersion and the crests still race ahead at v_p while the information stays with the envelope at v_g.

Worked examples

Example 1 — Free electron wave packet.

β = d²ω/dk² = d²(ℏk²/2m)/dk² = ℏ/m

Free-particle dispersion ω = ℏk²/2m; differentiate twice. A

β = (1.055×10⁻³⁴ J s)/(9.11×10⁻³¹ kg) = 1.16×10⁻⁴ m² s⁻¹

Insert ℏ and the electron mass; units J s / kg = m² s⁻¹. A

τ = 2σ₀²/β = 2(1.0×10⁻⁹ m)²/(1.16×10⁻⁴ m² s⁻¹) = 1.73×10⁻¹⁴ s

Spreading time for an initial width σ₀ = 1 nm. B

at t = 1 ps: βt/2σ₀² = t/τ = 10⁻¹²/1.73×10⁻¹⁴ = 57.9

Evaluate the dimensionless spreading parameter. B

σ(t) = σ₀√(1 + 57.9²) ≈ 57.9 σ₀ = 57.9 nm

Long-time regime (t ≫ τ), so σ ≈ σ₀·(t/τ). C

σ(1 ps) ≈ 58 nm from σ₀ = 1 nm

Reading. A 1 nm electron packet delocalizes to ~58 nm in a picosecond — a 58-fold spread — confirming why electron localization is so fragile. Spreading time τ ≈ 17 fs.

Example 2 — Deep-water gravity waves (ocean swell).

ω = √(gk), v_g = dω/dk = ½√(g/k), β = d²ω/dk² = −¼√(g/k³)

Deep-water dispersion relation; differentiate once for v_g, twice for β. B

k₀ = 0.10 rad m⁻¹ (λ ≈ 63 m), g = 9.81 m s⁻²

Choose a representative swell wavenumber. A

v_p = √(g/k₀) = √98.1 = 9.90 m s⁻¹, v_g = ½v_p = 4.95 m s⁻¹

Phase and group speeds; the famous factor of ½ for deep water. B

β = −¼√(9.81/0.001) = −¼(99.0) = −24.8 m² s⁻¹

Evaluate the curvature at k₀; sign is irrelevant to width since only β² enters. B

σ₀ = 50 m, t = 300 s: |β|t/2σ₀² = 24.8×300/(2×2500) = 1.49

Dimensionless spreading parameter for a 5-minute run. B

σ(t) = 50√(1 + 1.49²) = 50√3.21 = 89.6 m, centre at v_gt = 1485 m

Apply the width law and the group-velocity translation. C

σ(300 s) ≈ 90 m, envelope travels ~1.5 km at 4.95 m s⁻¹

Reading. The swell group moves at half the crest speed and nearly doubles in length over five minutes. Because v_g < v_p, individual crests appear to rise at the back of the group, march through it, and vanish at the front.

Problems

(A) Light in vacuum obeys ω = ck with c = 3.0×10⁸ m s⁻¹. Find v_p, v_g and β, and state what happens to a pulse of width σ₀ = 1 μm after 1 km.
Solution
v_p = ω/k = c = 3.0×10⁸ m s⁻¹. v_g = dω/dk = c = 3.0×10⁸ m s⁻¹. β = d²ω/dk² = 0. Since β = 0, σ(t) = σ₀ = 1 μm for all times — the pulse propagates 1 km (in t = 3.3 μs) with no change of shape. Vacuum is perfectly non-dispersive.
(B) An electron packet starts at σ₀ = 0.5 nm. How long until its width doubles? (β = ℏ/m = 1.16×10⁻⁴ m² s⁻¹.)
Solution
Doubling means σ(t) = 2σ₀, so 1 + (βt/2σ₀²)² = 4, giving βt/2σ₀² = √3. Hence t = 2√3 σ₀²/β = 2(1.732)(0.5×10⁻⁹)²/(1.16×10⁻⁴) = 3.464×2.5×10⁻¹⁹/1.16×10⁻⁴ = 7.5×10⁻¹⁵ s ≈ 7.5 fs.
(B) For a deep-water wave with k₀ = 0.05 rad m⁻¹ (g = 9.81 m s⁻²), find v_p, v_g, the ratio v_g/v_p, and |β|.
Solution
v_p = √(g/k₀) = √(9.81/0.05) = √196.2 = 14.0 m s⁻¹. v_g = ½v_p = 7.0 m s⁻¹, so v_g/v_p = ½ (universal for deep water). |β| = ¼√(g/k₀³) = ¼√(9.81/1.25×10⁻⁴) = ¼√78480 = ¼(280.1) = 70.0 m² s⁻¹.
(C) A "large" electron packet has σ₀ = 2 nm. Compute its width after t = 10 fs and compare with the 1 nm packet of Example 1. (β = 1.16×10⁻⁴ m² s⁻¹.)
Solution
βt/2σ₀² = (1.16×10⁻⁴)(10⁻¹⁴)/(2×(2×10⁻⁹)²) = 1.16×10⁻¹⁸/(8×10⁻¹⁸) = 0.145. σ = 2√(1 + 0.145²) = 2√1.021 = 2.02 nm — only 1% broadening. The 1 nm packet at the same time has t/τ = 10⁻¹⁴/1.73×10⁻¹⁴ = 0.58, giving σ = 1√(1+0.58²) = 1.16 nm, a 16% broadening. The larger packet spreads far more slowly, as τ ∝ σ₀².
(C) Define the spreading time τ = 2σ₀²/β. For two free-electron packets with σ₀ = 1 nm and σ₀ = 10 nm, compute both τ and their ratio, and interpret. (β = 1.16×10⁻⁴ m² s⁻¹.)
Solution
τ(1 nm) = 2(10⁻⁹)²/1.16×10⁻⁴ = 2×10⁻¹⁸/1.16×10⁻⁴ = 1.73×10⁻¹⁴ s ≈ 17 fs. τ(10 nm) = 2(10⁻⁸)²/1.16×10⁻⁴ = 2×10⁻¹⁶/1.16×10⁻⁴ = 1.73×10⁻¹² s ≈ 1.7 ps. Ratio = (10/1)² = 100, since τ ∝ σ₀². Interpretation: to keep a packet localized for 100× longer, make it only 10× wider — tight localization is bought at a steep price in spreading rate, the classic time–frequency (uncertainty) trade-off.