Chapter 07.01 Phase vii 24 / 57

The free particle

A Gaussian packet spreads under H = p²/2m

Wavepacket Dynamics 1 of 3 in phase 12 min read

A diffuse Gaussian wavepacket drawn as a soft cobalt bell curve, smearing rightward across a dark background, with thin horizontal carrier ripples visible inside the envelope. — Image: Wikimedia Commons · CC BY-SA 4.0 · Becarlson

In 1927, while the founders of the new mechanics were still arguing about what their equations meant, Charles Galton Darwin (grandson of the naturalist) sat down with Schrödinger's wave equation and asked the simplest possible question: what does a free electron actually do? The answer was startling. A perfectly Gaussian packet, left alone in empty space, slowly comes apart. It moves, yes, but it also smears. Within a billionth of a second an electron localized to atomic size has spread across an entire chip.

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Phase vii · Wavepacket Dynamics · Chapter 01

The free particle

The phrase “free particle” sounds like the easiest possible problem in quantum mechanics. No walls, no potential, no electric field, no neighbours. Just an electron drifting in vacuum. In classical physics it is the easiest problem: Newton says the velocity is constant, the position is a straight line, and a child with a ruler can predict where the particle will be a second from now. In quantum mechanics it is also the easiest problem, in the sense that the math has a closed form. But the answer it returns is not a straight line. It is a cloud that walks and a cloud that grows. That growing is the central, slightly unsettling fact this chapter wants to lodge in your bones.

The setup is minimal. Take the Schrödinger equation, set the potential to zero, and you are left with the kinetic-energy term alone. The Hamiltonian is H = p²/(2m). Plug it into the time-dependent equation and you have iℏ ∂ψ/∂t equal to (−ℏ²/2m) ∂²ψ/∂x². The right-hand side is a second derivative in space; the left-hand side is a first derivative in time. That asymmetry is the whole story. It is the reason a packet spreads, the reason “free” does not mean “boring,” and the reason an atom’s electron, were you to switch off the proton, would not sit politely where you found it.

Why does the asymmetry matter? Because plane waves of different momenta are solutions of this equation, and each one ticks at its own clock. A plane wave with wavenumber k has the form exp(ikx − iω(k)t), and the equation demands ω(k) = ℏk²/(2m). The frequency is quadratic in k, not linear. That sentence has more consequences than it has letters. If you build a localized packet by adding plane waves together (which you must, because a single plane wave is spread across all of space), every component evolves with a slightly different phase rate. The high-momentum pieces sprint ahead. The low-momentum pieces lag. The packet keeps its centre roughly where Newton would have put it, but the carrier waves underneath get out of step. The bell curve broadens.

History Darwin's quiet calculation

The first person to write down the spreading explicitly was Charles Galton Darwin in 1927, not in a treatise but in a short paper published in the Proceedings of the Royal Society. He wanted to check whether Schrödinger’s ψ could honestly represent a real moving electron. He set up a Gaussian at t = 0, plugged it into the free equation, and ground out the integral. The width of the packet at time t, he found, came out to be the initial width and a new term proportional to ℏt over the mass. He wrote, drily, that this was “a phenomenon which has no parallel in classical mechanics.” Later that same year Paul Ehrenfest noticed the time for a matter wave to double its width is roughly m(Δx)²/ℏ, a formula now hung on every introductory blackboard. Neither result made the front pages. Both became the daily bread of every quantum optics lab.

The Gaussian is the natural shape to study because it lives on the floor of the uncertainty principle. Last phase you saw the inequality Δx · Δp ≥ ℏ/2, and you saw that the Gaussian was the unique shape that saturates the bound: for any other wavefunction the product is strictly larger. So take a Gaussian centred at the origin with width σ in position. Its Fourier transform is another Gaussian, centred on some momentum p₀, with width ℏ/(2σ) in momentum. The narrower you squeeze the position bell, the wider the momentum bell must be, and at the moment t = 0 you have packed both as tightly as nature permits.

What happens next is a competition between the momentum spread you cannot remove and the time you give it to work. Each plane wave inside the packet evolves as exp(−iE(p)t/ℏ), and E(p) = p²/(2m) is the kinetic energy. That means the phase rate is p²t/(2mℏ), and crucially, different p values have different phase rates. After a time t the carrier waves that were locked together at t = 0 have drifted out of step by an amount proportional to (Δp)²·t. Once that drift exceeds 2π the components no longer add up constructively in a single tight spot. They reconstruct over a wider region. The packet is wider.

The formula for the new width is clean. If σ₀ is the initial position spread, then at time t the width has grown to σ(t) = σ₀ · √(1 + (ℏt/(2mσ₀²))²). At early times the second term inside the square root is tiny, and the packet barely notices the passage of time. At late times the second term dominates, and the width grows linearly with t at the rate ℏ/(2mσ₀). That late-time growth rate is exactly the velocity uncertainty Δp/m = (ℏ/(2σ₀))/m. The packet spreads at the speed it is uncertain about its own momentum. There is a poetic symmetry there: the packet you knew most precisely is the packet that flies apart the fastest.

In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an envelope. A wave packet can be analyzed into, or can be synthesized from, a potentially-infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Any signal of a limited width in time or space requires many…

From Wikipedia, “Wave packet” — https://en.wikipedia.org/wiki/Wave_packetCC BY-SA 4.0

Now look at how the packet moves. The centre of mass travels at the group velocity, the speed at which the envelope of the packet drifts. For our dispersion relation ω(k) = ℏk²/(2m), the group velocity is dω/dk = ℏk₀/m, which is just p₀/m. That is exactly the classical velocity you would have written down: momentum divided by mass. The centre of the packet obeys Newton’s first law. So far so reassuring. The classical particle is hiding inside the quantum description, riding at the centre of the bell curve.

But the carrier wave inside the envelope, the rapid ripple, moves at a different speed. Its phase velocity is ω/k, which is ℏk₀/(2m) = p₀/(2m). That is half the group velocity. Stare at a packet for a second and you will see the envelope move forward at the classical speed while the ripples beneath it slide backwards relative to the envelope, at the same speed. Like watching the spokes of a wheel in a film: the envelope is doing one thing, the substructure is doing another. The classical particle is the envelope. The quantum substrate is the carrier. They do not move at the same speed, and they cannot.

Story Schrödinger's hope, Schrödinger's disappointment

Schrödinger, when he discovered his wave equation in 1926, had a dream. He hoped wavepackets would be the long-sought reconciliation of waves and particles: a tightly localized blob of ψ, drifting through space along a Newtonian trajectory, would just be the electron. No collapse, no probability, no philosophical headache. He worked out a beautiful case, the coherent state of the harmonic oscillator, that does exactly this: it oscillates without spreading. He was elated. Then Heisenberg’s paper on uncertainty appeared, and within weeks Heisenberg pointed out (politely, in print) that Schrödinger’s trick only worked for the harmonic oscillator. For a free particle, or for an electron in the Coulomb potential of an atom, the packet must spread. Schrödinger, in private letters, called this discovery “depressing.” It was. The wavefunction was not the particle. The probability interpretation Born had been pushing all year was the only consistent reading. The wavepacket dream was dead by Christmas.

The free Gaussian packet at four snapshots in time. The envelope drifts rightward at the group velocity v = p₀/m. As it travels, it broadens and flattens, because the higher-k components race ahead and the lower-k components lag. Area under each curve is conserved (probability), but the support widens monotonically.

There is a way to picture the two velocities at once that physicists have used since Rayleigh wrote his Theory of Sound in 1877. Plot the dispersion relation ω(k) against k. For a stretched string, ω is proportional to k and the plot is a straight line through the origin; phase velocity (the slope of the line from origin to point) and group velocity (the slope at the point) are equal, and the packet keeps its shape forever. For a free quantum particle, ω(k) = ℏk²/(2m) is a parabola. The slope from origin to the carrier point k₀ gives the phase velocity ℏk₀/(2m). The slope of the tangent at k₀ gives the group velocity ℏk₀/m. The tangent is twice as steep as the chord. Group is twice phase. The envelope outruns its own ripple, every time.

The dispersion relation of a free quantum particle is a parabola, not a straight line. Phase velocity is the slope of the chord from the origin to the carrier point k₀; group velocity is the slope of the tangent at the same point. The tangent is twice as steep. The envelope runs at the classical speed p/m. The carrier slips backward inside it at exactly half that rate.

Run the 2D wavepacket sandbox in 'free' mode. Set V = 0 everywhere and watch the Gaussian drift and broaden. Pinch it tight and watch it explode; spread it wide and watch it crawl. The math is the same; the mass is the same; only your initial σ changes.open ↗

Derive the spreading formula σ(t) = σ₀ √(1 + (ℏt/(2mσ₀²))²)

Take an initial Gaussian, centred at x = 0 with width σ₀ and average momentum ℏk₀:

ψ(x, 0) = (2π σ₀²)^(-1/4) · exp[ -x²/(4σ₀²) + i k₀ x ]

Write it as a Fourier sum of plane waves with amplitudes φ(k):

φ(k) = (2σ₀² / π)^(1/4) · exp[ -σ₀² (k − k₀)² ]

The amplitude is itself a Gaussian in k, centred on k₀ with width 1/(2σ₀). That is the momentum-space Gaussian; its widths in x and k saturate Δx · Δk = 1/2, so Δx · Δp = ℏ/2.

Each plane wave evolves trivially under the free Hamiltonian:

exp(ikx) → exp[ i(kx − ω(k) t) ],   ω(k) = ℏk² / (2m)

So at time t the full wavefunction is

ψ(x, t) = ∫ φ(k) · exp[ i(kx − ℏk² t / (2m)) ] dk

The integral is Gaussian in k, with a complex coefficient. Complete the square in k, do the standard Gaussian integral, and the result is

ψ(x, t) ∝ (σ₀² + iℏt/(2m))^(-1/2) · exp[ -(x − v₀ t)² / (4(σ₀² + iℏt/(2m))) + i(k₀ x − ω₀ t) ]

where v₀ = ℏk₀/m is the group velocity. Take the probability density |ψ(x, t)|². The imaginary part in the denominator combines with its complex conjugate to give

|ψ(x, t)|² ∝ exp[ -(x − v₀ t)² / (2 σ(t)²) ],   σ(t)² = σ₀² + (ℏt / (2mσ₀))²

σ(t) = σ₀ · √( 1 + (ℏt / (2mσ₀²))² )

Two limits worth pocketing. At early times, ℏt ≪ 2mσ₀², the width is barely changed. At late times, ℏt ≫ 2mσ₀², the width grows linearly: σ(t) ≈ ℏt / (2mσ₀) = (Δp/m) · t. The growth rate equals the velocity uncertainty Δv = Δp/m, which is the picture you would have drawn classically if you had been told the particle’s velocity was uncertain by that much.

Ehrenfest’s doubling time, σ(T) = 2σ₀, comes from solving 1 + (ℏT/(2mσ₀²))² = 4. So T = √3 · 2mσ₀²/ℏ, or to a slogan’s accuracy T ≈ m σ₀² / ℏ. For an electron at σ₀ = 1 Å this is around 10⁻¹⁶ s. For a 1 µg dust grain at σ₀ = 1 µm, T is about 10²² s, longer than the age of the universe. Quantum spreading is real for everything. It only matters at small scales.

Gotcha The packet is not the particle, the |ψ|² is the probability

Schrödinger spent the rest of his career arguing for a literal reading of the wavepacket as the particle itself. He lost. The standard reading, the one every modern experiment confirms, is Born’s: |ψ(x,t)|² is the probability density to find the particle at x at time t. When the packet spreads, the particle is not actually “smearing” in any classical sense. What is spreading is the probability distribution over where a position measurement would, if performed, find a single localized hit. Run the same prepared state through a screen a thousand times and the hits will trace the bell curve. Each individual hit is sharp. Quantum spreading is statistical, not material. The Hollywood image of an electron melting into a fog is wrong in a specific way: the electron does not melt, the bookkeeping does.

The spreading of free wavepackets is not a curiosity. It controls things you might not have linked together. The linewidth of a freely radiating atom can be traced to the limited lifetime of its excited state, which is itself an energy-time uncertainty cousin of the spreading we just derived. The diffraction pattern of a slow neutron through a crystal is set by the spread of the neutron’s wavepacket compared to the lattice spacing. The resolution of an electron microscope is bounded by how tightly you can launch an electron and how far it must travel before the wave geometry blurs. Every modern semiconductor device, where electrons hop between traps a few nanometres apart, is alive only because the electron’s wavepacket is wider than the gap. The spreading is not the failure of localization. It is what makes quantum tunnelling possible, what gives matter waves their reach, what allows you to detect a particle on a screen that is far away from the slit that prepared it.

There is one last thing worth saying. The wavepacket spreads because the dispersion is parabolic, because ω is not linear in k. In other waves, light in vacuum, sound in air, the dispersion is linear and packets keep their shape forever. (Light pulses keep their shape over interstellar distances because Maxwell’s vacuum is non-dispersive; sound from a far-off thunderclap stays a thunderclap because air is roughly non-dispersive at audible frequencies.) Quantum matter is not like that. The Schrödinger equation has ω ∝ k², the result is dispersive, and so any free packet must change shape as it moves. The classical particle, that little marble travelling on a straight line, is what you see when the packet is wide enough that its spreading happens too slowly to matter at the human timescale. The marble is a slow-moving Gaussian whose width happens to be far below what your eye can resolve. The particle picture is the limit of the wave picture. The wave picture is the truth.

You have now watched a single Gaussian come apart in empty space. Next we run the same packet at two open slits at once and let it interfere with itself. The spreading we just derived is what makes the screen pattern possible. Every dark fringe is a place the packet refuses to land.

next chapter → The double slit, revisited