Wave packets: phase and group velocity

Take a single de Broglie wave, ψ = A e^{(i(kx − ωt))}. Its modulus squared is |A|² at every x and every t, so the particle is equally likely here, in the next room, and outside the galaxy, and no feature of the wave ever changes that. It is not even normalisable — ∫|A|² dx diverges — so strictly it is not a state, only a convenient limit. The repair is to add a band of them, ψ(x, t) = (1/√(2π)) ∫ φ(k) e^{(i(kx − ω(k)t))} dk, with φ(k) peaked at some k₀ and of width Δk. The components interfere constructively across a region of width Δx ≈ 1/Δk and cancel outside it, and that lump is the thing you can point at. Nothing has been added to the theory; the rest of this lesson is simply what that integral does as t runs on.

The integral is identical for light, sound, ripples and electrons. Only ω(k) differs, and ω(k) is where the physics enters. For a free non-relativistic particle, E = p²/2m with E = ħω and p = ħk gives ω(k) = ħk²/2m, a parabola through the origin. For light in vacuum, ω = ck, a straight line. That difference decides everything. On a straight line through the origin, ω/k and dω/dk are the same number, so light's crests and envelope both travel at c and a vacuum pulse keeps its shape forever. On a parabola, ω/k = ħk/2m, dω/dk = ħk/m, and the second derivative ħ/m is not zero — so a matter packet has two different speeds and cannot keep its shape. At k₀ = 1.0 × 10¹⁰ m⁻¹: ω = 5.79 × 10¹⁵ rad s⁻¹, ω/k = 5.79 × 10⁵ m s⁻¹, dω/dk = 1.16 × 10⁶ m s⁻¹.

Expand about the centre of the band: ω(k) ≈ ω₀ + ω′(k₀)(k − k₀) + ½ω″(k₀)(k − k₀)². Pull out the carrier e^{(i(k₀x − ω₀t))} and what remains, to first order, depends on x and t only through the combination x − ω′(k₀)t: the envelope keeps its shape and slides at v_g = ω′(k₀). Equivalently, the components add up only where their phases are stationary in k, and that condition is x = ω′(k₀)t. Two waves show it without calculus. Superpose k₁ = 0.95 × 10¹⁰ m⁻¹ and k₂ = 1.05 × 10¹⁰ m⁻¹ for a free electron: ω₁ = 5.226 × 10¹⁵ and ω₂ = 6.385 × 10¹⁵ rad s⁻¹, so the envelope runs at Δω/Δk = 1.16 × 10⁶ m s⁻¹ while the carrier runs at ω̄/k̄ = 5.81 × 10⁵ m s⁻¹ — a ratio of 1.995, heading for exactly 2 as the band narrows.

Potential energy has no absolute zero: shift it by V₀ and no force and no measurement changes. In the wave, that shift multiplies ψ by e^{(−iV₀t/ħ)}, a global phase, so |ψ|² is untouched at every x and t. But it also sends ω(k) → ω(k) + V₀/ħ, which leaves dω/dk exactly alone and moves ω/k by V₀/(ħk). Adding just 10 eV to the zero lifts the phase velocity of that 3.81 eV electron from 5.79 × 10⁵ to 2.10 × 10⁶ m s⁻¹, while its group velocity sits still. A quantity a bookkeeping choice can rescale is not the speed of anything. Relativity says it louder: with E = γmc², ω/k = E/p = c²/v exceeds c for every massive particle, while dω/dk = dE/dp = pc²/E = v is the speed you measure.

Keep the second-order term and the envelope stops moving rigidly. For a Gaussian packet of initial rms width σ₀, the exact free solution gives σₓ(t) = σ₀√(1 + (t/τ)²) with τ = 2mσ₀²/ħ, while the momentum spread Δp = ħ/2σ₀ never changes — the components that were there from the start simply run apart at Δv = ħ/2mσ₀. For an electron with σ₀ = 1.0 nm, τ = 17 fs and Δv = 5.79 × 10⁴ m s⁻¹, so after 1.0 ns its rms width has grown to 58 µm. Narrow packets die fastest, since τ goes as σ₀²; heavy ones barely spread, since τ goes as m, and a 1.0 nm proton packet needs 32 ps to reach √2 σ₀. That mass scaling is a large part of why the classical limit looks classical.

The band is not optional. Δx Δk ≥ ½ holds for any square-integrable packet, so pinning an electron to σ₀ = 1.0 nm demands Δk ≥ 5 × 10⁸ m⁻¹ and therefore Δp ≥ ħ/2σ₀ = 5.3 × 10⁻²⁶ kg m s⁻¹. Those are the very components that make it spread, so the sharper the packet the shorter it lasts: the two statements are one statement. What the packet does not do is swell as a lump of charge. |ψ|² is a probability density for one electron, and a screen 1.0 ns downstream still records one point-like arrival — only its position is now less predictable. Nor is the growth unconditional: a packet prepared converging narrows first, reaches a minimum, and only then spreads without bound.