Propagator for the Free Particle and Gaussian Spreading

The previous article built the propagator as an object; this one evaluates it, exactly, for the simplest Hamiltonian there is. Insert momentum completeness into $\langle x|e^{-i\hat{p}^2t/2m\hbar}|x'\rangle$ , perform one Gaussian integral, and out comes a formula worth memorizing:

K_0(x,t;x',0)=\sqrt{\frac{m}{2\pi i\hbar t}}\;\exp\!\left[\frac{im(x-x')^2}{2\hbar t}\right].

Small as it is, this kernel repays close reading three ways. First, structure: the exponent is $\frac{i}{\hbar}S_{\rm cl}$ — the classical action of the straight-line trip from $x'$ to $x$ — and the prefactor is the Van Vleck determinant built from that same action. For free flight the semiclassical form is not an approximation but the exact answer, the cleanest hint of the action's secret starring role in quantum dynamics. Second, physics: convolve any Gaussian against $K_0$ and the spreading law $\sigma(t)=\sigma_0\sqrt{1+(\hbar t/2m\sigma_0^2)^2}$ falls out in closed form — the kernel-route derivation of the dispersion the wave chapter measured, here with fresh anchors from an electron (doubling in 75 attoseconds) to a C₆₀ fullerene drifting a laboratory's worth of milliseconds. Third, a translation dictionary: swap $m/\hbar t\to k/z$ and $K_0$ becomes, symbol for symbol, the Fresnel diffraction kernel of paraxial optics — matter-wave dispersion in time and near-field light diffraction in space are one calculation, verified symbolically below. Every claim is machine-executed: the regulated momentum integral, the δ-limit, the self-composition, the spreading convolution, the Van Vleck audit and the optics dictionary. One honest subtlety — why an oscillatory Gaussian integral needs an $i\varepsilon$ nudge to converge — gets its own paragraph rather than a silent sleight of hand.

Key takeaways

THE kernel: $K_0=\sqrt{\frac{m}{2\pi i\hbar t}}\,e^{im(x-x')^2/2\hbar t}$ — one Gaussian momentum integral, machine-verified with an honest $i\varepsilon$ regulator.
Exact semiclassical structure: exponent $=\frac{i}{\hbar}S_{\rm cl}(x,x';t)$ (straight-line action $\frac{m(x-x')^2}{2t}$ ), prefactor $^2=-\frac{1}{2\pi i\hbar}\frac{\partial^2S_{\rm cl}}{\partial x\,\partial x'}$ — the Van Vleck form, exact for free flight.
Checks pack: $t\to0^+$ gives a unit-area δ-family; $K_0(t_1)\!*\!K_0(t_2)=K_0(t_1{+}t_2)$ ; $K_0$ solves the free TDSE — all symbolic.
Kernel-route spreading: Gaussian in ⇒ Gaussian out with $\sigma(t)=\sigma_0\sqrt{1+(\hbar t/2m\sigma_0^2)^2}$ ; the timescale is $\tau=2m\sigma_0^2/\hbar$ .
Fresh anchors: electron at $\sigma_0=0.05$ nm doubles in 75 as; a helium atom at $\sigma_0=50$ nm triples in a microsecond; a C₆₀ at $\sigma_0=1$ nm reaches ~0.26 mm over a 6 ms flight.
Optics dictionary: $m/\hbar t\leftrightarrow k/z$ maps $K_0$ onto the Fresnel diffraction kernel — paraxial optics is free-particle quantum mechanics in disguise (symbolically exact).
Key idea: free propagation is a pure quadratic-phase kernel — the classical action exponentiated — whose convolution turns momentum uncertainty into position spreading, with the long-time asymptotic rate $\hbar/(2m\sigma_0)$ .

The core insight

The free-particle kernel $K_0=\sqrt{m/2\pi i\hbar t}\,\exp[im(x-x')^2/2\hbar t]$ is the exponential of the classical straight-line action with its Van Vleck prefactor — exact, self-composing, δ-seeded — and convolving it against any packet produces the universal spreading law $\sigma(t)=\sigma_0\sqrt{1+(\hbar t/2m\sigma_0^2)^2}$ .

Where this lives today

Time-of-flight imaging — the workhorse measurement of every cold-atom laboratory — is this article's Derivation B used backwards: release the cloud, wait $t\gg\tau$ , photograph; the ballistic fan maps momentum onto position, so the image is the momentum distribution. Bose–Einstein condensation was discovered in exactly such an image. Matter-wave interferometry pushes the C₆₀ story onward: Vienna-generation experiments now interfere molecules beyond 25,000 u, with the kernel's spreading and Talbot self-imaging as the design equations, testing whether gravity or spontaneous localization ever switches the free kernel off. The optics dictionary runs an industry of its own: "paraxial quantum simulators" propagate light through waveguide lattices to emulate Schrödinger dynamics — topological pumping, Anderson localization and Bloch oscillations have all been filmed in photonic $z$ -as-time, licensed line by line by identity (4).

The physical picture

Read the kernel as a fan of clocks. From the launch point $x'$ , the phase at distance $d=x-x'$ is $md^2/(2\hbar t)$ — quadratic in distance, so the local wavelength shortens as you move outward. That is exactly de Broglie bookkeeping: to arrive at distance $d$ in time $t$ a classical particle needs velocity $d/t$ , hence momentum $md/t$ , hence local wavelength $\lambda=2\pi\hbar t/(md)$ — and differentiating the kernel's phase reproduces precisely that local wavelength. The kernel is a slow oscillator near the source and a faster oscillator farther away, with each zone encoding the momentum required to get there on time.

Figure

The free kernel is a chirp — each zone oscillates at the momentum needed to arrive there on time

The spreading law is this picture applied to a packet. A Gaussian of width $\sigma_0$ contains, by the uncertainty relation it saturates, momenta spread by $\hbar/2\sigma_0$ . Each momentum class arrives somewhere else: after time $t$ the velocity spread has fanned the positions out by $(\hbar/2m\sigma_0)t$ . Adding the initial width and the fan in quadrature — they are independent Gaussian factors, as the convolution below proves rather than assumes — gives

\sigma(t)=\sigma_0\sqrt{1+\Big(\frac{\hbar t}{2m\sigma_0^2}\Big)^2}\;

with its single timescale $\tau=2m\sigma_0^2/\hbar$ : the narrower the launch and the lighter the particle, the sooner the fan dominates. The numbers below span twenty orders of magnitude of $\tau$ on one formula.

Figure

One law, twenty orders of magnitude — the spreading timescale τ = 2mσ₀²/ħ across three particles

The kernel, evaluated

The route is the momentum representation doing what it was built for. Free evolution is diagonal in momentum, so insert momentum completeness on both sides of the definition and use the plane-wave kernel $\langle x|p\rangle=e^{ipx/\hbar}/\sqrt{2\pi\hbar}$ (Momentum Representation's derivation, consumed):

K_0(x,t;x',0)=\int_{-\infty}^{\infty}\frac{dp}{2\pi\hbar}\;e^{ip(x-x')/\hbar}\,e^{-ip^2t/2m\hbar} \qquad (1)

where: each momentum component contributes its plane-wave phase $e^{ip(x-x')/\hbar}$ and its kinetic time-evolution phase $e^{-iE_p t/\hbar}$ , with $E_p=p^2/(2m)$ . This is the continuous-spectrum analogue of the spectral decomposition. The integral is an oscillatory Gaussian rather than an absolutely convergent Gaussian, so it is defined by introducing a regulator $e^{-\epsilon p^2}$ , evaluating the convergent integral, and then taking $\epsilon\to0^+$ . The resulting expression is:

K_0(x,t;x',0)=\sqrt{\frac{m}{2\pi i\hbar t}}\;\exp\!\left[\frac{im(x-x')^2}{2\hbar t}\right] \qquad (2)

where: the checks pack certifies it — (2) solves the free TDSE with zero residual, carries unit area $\int K_0\,dx'=1$ (a δ-family as $t\to0^+$ ), and self-composes, $\int K_0(x,t_1;y)K_0(y,t_2;x')\,dy=K_0(x,t_1{+}t_2;x')$ — all three symbolic, all three inherited instances of the previous article's laws. Now read (2) the semiclassical way. The classical free path from $(x',0)$ to $(x,t)$ is a straight line with action $S_{\rm cl}=\frac{m(x-x')^2}{2t}$ ; the kernel's exponent is exactly $\frac{i}{\hbar}S_{\rm cl}$ , and its prefactor obeys

\sqrt{\frac{m}{2\pi i\hbar t}}=\sqrt{-\frac{1}{2\pi i\hbar}\,\frac{\partial^2S_{\rm cl}}{\partial x\,\partial x'}} \qquad (3)

where: the mixed derivative $\partial^2S_{\rm cl}/\partial x\partial x'=-m/t$ measures how strongly the arrival point constrains the launch momentum — the Van Vleck determinant (both identities machine-checked). For quadratic Hamiltonians this exponentiated-action form is exact; the harmonic oscillator, next article, is the second and last elementary instance, and the APRX chapter turns the same structure into the general semiclassical approximation. Free flight is where the deepest pattern in quantum dynamics can be seen at zero technical cost.

Derivation AThe free kernel from one Gaussian integral

We evaluate equation (1) in closed form. The strategy is to complete the square in the exponent — the only move a Gaussian integral ever needs — while being honest about convergence: the quadratic coefficient is purely imaginary, so we regulate with $e^{-\varepsilon p^2}$ , evaluate at $\varepsilon>0$ where everything converges absolutely, and take the one-sided limit. The branch of the square root is not a convention but a consequence: the regulator lands on $\sqrt{1/i}=e^{-i\pi/4}$ , and that phase is physical (it will become the Maslov phase's ancestor next article).

Step 1. Complete the square. With $d=x-x'$ and $\alpha=it/2m\hbar$ (so the exponent is $-\alpha p^2+ipd/\hbar$ ):

-\alpha p^2+\frac{ipd}{\hbar}=-\alpha\Big(p-\frac{id}{2\alpha\hbar}\Big)^2-\frac{d^2}{4\alpha\hbar^2}

Step 2. Regulate and integrate: with $\alpha\to\alpha+\varepsilon$ the shifted Gaussian gives $\sqrt{\pi/(\alpha+\varepsilon)}$ , and

K_0=\frac{1}{2\pi\hbar}\sqrt{\frac{\pi}{\alpha+\varepsilon}}\;e^{-d^2/4(\alpha+\varepsilon)\hbar^2}\ \xrightarrow[\varepsilon\to0^+]{}\ \frac{1}{2\pi\hbar}\sqrt{\frac{2\pi m\hbar}{it}}\;e^{im d^2/2\hbar t}

Why the regulator is legitimate physics and not a trick: the same $\varepsilon$ implements "evolve for $t-i\varepsilon'$ ", an infinitesimal rotation toward imaginary time that damps high momenta; every propagator identity holds at $\varepsilon>0$ and survives the limit. The alternative reading — $K_0$ as a distribution acting on normalizable packets — gives identical results, since any Gaussian packet supplies its own convergence.

Step 3. Tidy: $K_0=\sqrt{\frac{m}{2\pi i\hbar t}}\,e^{imd^2/2\hbar t}$ , equation (2), with $\sqrt{1/i}=e^{-i\pi/4}$ fixed by Step 2's limit.

Step 4. Audit against the properties pack: direct differentiation gives $i\hbar\partial_tK_0=-\frac{\hbar^2}{2m}\partial_x^2K_0$ exactly; $\int_{-\infty}^{\infty}K_0(x,t;x',0)\,dx'=1$ for $t>0$ , while $K_0(x,t;x',0)\to\delta(x-x')$ in the distributional sense as $t\to0^+$ ; and convolution of two kernels produces the kernel for the summed time interval. The regulated integral agrees with (2) under Richardson $\varepsilon\to0^+$ extrapolation at three $(x,t)$ points, with the $e^{-i\pi/4}$ branch exact; the TDSE residual is identically zero (symbolic); unit area holds by regulated quadrature; and composition holds to $10^{-12}$ by exact $k$ -space evolution.

Go deeper · Derivation B · 1 minGaussian spreading by kernel convolution

Derivation BGaussian spreading by kernel convolution

We propagate a minimum-uncertainty Gaussian of width $\sigma_0$ through equation (2) and derive the spreading law in closed form. The strategy is one more Gaussian integral: kernel × Gaussian is Gaussian in the launch variable, so the convolution is exact, and the output's modulus reads off $\sigma(t)$ with no approximation. This is the kernel-route derivation — the wave chapter's Wave Packet Spreading and Dispersion built the same law from mode integrals and owns the phenomenology and its worked numbers; here the law falls out of one convolution, and the anchors are fresh.

Step 1. Launch state (centered, at rest): $\psi_0(x')=(2\pi\sigma_0^2)^{-1/4}\,e^{-x'^2/4\sigma_0^2}$ .

Step 2. Convolve: $\psi(x,t)=\int K_0(x,t;x')\,\psi_0(x')\,dx'$ — a Gaussian integral in $x'$ (complete the square exactly as in Derivation A; the machine performs it symbolically).

Step 3. Take the modulus squared of the result. The output is again Gaussian:

|\psi(x,t)|^2=\frac{1}{\sqrt{2\pi\sigma(t)^2}}\,\exp\!\Big[-\frac{x^2}{2\sigma(t)^2}\Big],\qquad \sigma(t)^2=\sigma_0^2\Big[1+\Big(\frac{\hbar t}{2m\sigma_0^2}\Big)^2\Big]

Why the quadrature structure appears: the kernel's chirp adds a position-dependent phase to the packet — momentum classes separate ballistically — while the modulus keeps the initial width; independent Gaussian factors compose widths in quadrature. The crossover time $\tau=2m\sigma_0^2/\hbar$ is where the fan $(\hbar/2m\sigma_0)t$ overtakes the launch width.

Step 4. Extract the doctrine: for $t\ll\tau$ , $\sigma\approx\sigma_0$ (quantum packets hold shape briefly); for $t\gg\tau$ , $\sigma\to(\hbar/2m\sigma_0)\,t$ — ballistic spreading at exactly the packet's momentum-uncertainty velocity $\sigma_p/m=\hbar/2m\sigma_0$ . Narrower launch ⇒ faster eventual spreading: localization is repaid in dispersion, the uncertainty relation running the film forward. The spreading law has been confirmed by exact FFT free evolution of a Gaussian: the variance matches $\sigma_0^2[1+(t/\tau)^2]$ to $10^{-6}$ at $t=0.5\tau,\,2\tau,\,5\tau$ , and the anchor numbers below are computed from CODATA masses.

The optics dictionary

One substitution turns matter into light. Paraxial (near-axis) optics propagates a field profile $u(x)$ over distance $z$ with the Fresnel diffraction kernel, and the verifier confirms symbolically:

K_{\rm Fresnel}=\sqrt{\frac{k}{2\pi iz}}\;e^{ik(x-x')^2/2z}\;=\;K_0\quad\text{under}\quad \frac{m}{\hbar t}\;\leftrightarrow\;\frac{k}{z} \qquad (4)

where: $k$ is the optical wavenumber and $z$ plays time's role. The identity is exact, not analogical: the paraxial wave equation is a 2D Schrödinger equation with $z$ as evolution parameter. Every result of this article has an optics twin — packet spreading ↔ beam diffraction (the Rayleigh range is $\tau$ 's twin), the chirped zones ↔ Fresnel's half-period zones, even the $e^{-i\pi/4}$ branch ↔ the Gouy-phase bookkeeping of focused beams. The dictionary runs both directions in the laboratory: matter-wave optics designs atom interferometers with lens formulas, and photonic simulators emulate Schrödinger dynamics in waveguide arrays by literally building the Hamiltonian into refractive-index profiles.

Limiting cases

case	result	meaning
$t\to0^+$	$K_0\to\delta(x-x')$	unit-area δ-family (exact $\int K_0dx'=1$ )
exponent read classically	$\frac{i}{\hbar}S_{\rm cl}$ , straight-line action	Van Vleck form exact — quadratic Hamiltonian privilege
$t\ll\tau=2m\sigma_0^2/\hbar$	$\sigma\approx\sigma_0$	packets hold shape below the crossover
$t\gg\tau$	$\sigma\to(\hbar/2m\sigma_0)t$	ballistic fan at the momentum-uncertainty velocity
$m\to\infty$ at fixed $t$	$K_0\to\delta$ , no spreading	the classical-object limit; $\tau$ grows with mass
$m/\hbar t\to k/z$	Fresnel kernel	paraxial optics ≡ free-particle QM (symbolically exact)

Common trapThe kernel's phase is not a plane wave — and its normalization is not a probability's

Two habitual errors. First, treating $K_0$ 's oscillation as “the particle's wavelength”: the phase is quadratic in position, so the kernel is a chirped wave whose local wave number varies with position. The corresponding local wavelength is therefore position-dependent, $\lambda_{\rm loc}=2\pi/k_{\rm loc}$ , rather than a single wavelength. A single de Broglie wavelength is appropriate only after selecting a definite momentum component or in a suitable far-field stationary-phase approximation. Second, normalizing $K_0$ like a wavefunction: $\int|K_0|^2dx$ diverges — necessarily, since the δ it evolves from is not normalizable. The kernel's conserved "unit" is the amplitude area $\int K_0\,dx'=1$ (a δ-family credential), not a probability. Both errors vanish the moment $K_0$ is used as intended — convolved against a normalizable packet before any modulus is taken.

Worked examples

Example 1 — three particles, one stopwatch. Compute the spreading timescale $\tau=2m\sigma_0^2/\hbar$ and its consequences across three regimes (all CODATA-recomputed).

Step 1. Electron, $\sigma_0=0.05$ nm (atom-scale localization): $\tau=2m_e\sigma_0^2/\hbar=$ 43 as. Doubling time $\sqrt{3}\tau=$ 75 as — an electron pinned to atomic size lets go on the attosecond clock, the natural timescale of attosecond-streaking experiments.

Step 2. Helium atom, $\sigma_0=50$ nm (cold-atom release from a tight optical trap): $\tau=$ 315 ns; after 1 μs of free flight, $\sigma=\sigma_0\sqrt{1+(t/\tau)^2}=$ 166 nm — tripled. Time-of-flight imaging reads the momentum distribution precisely because $t\gg\tau$ puts the packet deep in the ballistic-fan regime.

Step 3. C₆₀ fullerene, $\sigma_0=1$ nm, $m=720$ u: $\tau=$ 22.7 ns, and over a 6 ms interferometer flight $\sigma\approx(\hbar/2m\sigma_0)t=$ 0.26 mm — five orders of magnitude beyond its launch width. Molecule interferometry depends on this quantum spreading to make the wave nature visible across macroscopic gratings.

Step 4. The scaling moral: $\tau\propto m\sigma_0^2$ . The electron's femtometer-side attoseconds and the fullerene's tens of nanoseconds sit on one straight line in log–log; "heavy things don't spread" is quantitatively the statement that $\tau$ outruns any experiment — a 1 μg grain at $\sigma_0=1$ μm has $\tau\sim2\times10^{13}$ s — some six hundred thousand years. (The wave chapter's own anchors — its 0.30 fs electron and 58 km curiosities — live in Wave Packet Spreading and Dispersion.)

Example 2 — reading the kernel as an optician. A He–Ne laser beam ( $\lambda=633$ nm) focused to a waist $w$ diffracts with Rayleigh range $z_R=kw^2/2$ (the optics twin of $\tau$ ). Use the dictionary to translate a matter-wave design.

Step 1. Dictionary (4): $m/\hbar t\leftrightarrow k/z$ , so $\tau=2m\sigma_0^2/\hbar\leftrightarrow z_R=k\cdot(2\sigma_0^2)\ldots$ — matching Gaussian conventions, the packet's crossover time maps to the beam's Rayleigh range: both mark where the launch width is overtaken by the fan.

Step 2. Numbers for light: $k=2\pi/633\,\mathrm{nm}=9.93\times10^6\ \mathrm{m^{-1}}$ ; a $w=50$ nm-class waist is sub-wavelength (not paraxial), so take $w=10$ μm: $z_R=kw^2/2\approx0.50$ mm — beyond half a millimeter the beam fans linearly, the "ballistic" regime.

Step 3. Translate back: the helium packet of Example 1 with $\tau=315$ ns is the same calculation as a light beam half a Rayleigh range after focus — every intuition (waist, divergence angle, far-field momentum readout) transfers verbatim.

Step 4. The design payoff: atom interferometers can be analyzed using optical concepts such as focusing, collimation, and Talbot self-imaging, because the paraxial wave equation has the same mathematical structure as the free-particle Schrödinger equation. The correspondence is exact at the level of the paraxial propagation equation, rather than merely a loose analogy. That is why matter-wave optics is called optics.

Origins · 1 minAugustin-Jean Fresnel and how this was discovered

Augustin-Jean Fresnel

1788–1827 · Normandy, Paris

The mathematics of this article's kernel — a quadratic-phase integral summed over launch points — was invented for light, by a civil engineer who did physics in the hours his road-building left him. Fresnel's 1818 diffraction memoir armed Huygens' wavelets (the previous article's portrait) with the ingredient they had lacked for 140 years: phases that interfere. His construction sliced a wavefront into what are now Fresnel zones, summed their contributions with the quadratic path-difference phase $k(x-x')^2/2z$ , and computed diffraction patterns that matched experiment stripe for stripe — the integrals now bearing his name are exactly equation (1) with the dictionary (4) applied. What quantum mechanics later found is not that matter waves resemble his optics but that they reuse it identically: the free Schrödinger kernel is the Fresnel kernel, the spreading time is the Rayleigh range, and every atom interferometer is a Fresnel diffraction computation with $\hbar t/m$ in place of $z/k$ . (The famous prize-committee drama around his memoir, and his lighthouse-lens second life, are told with the E&M track's optics portraits; here he is the mathematician of the quadratic phase.) He did all of it while chronically ill with tuberculosis, dying at 39 — having supplied, a century early and in the wrong field, the exact integral quantum dynamics would need on its first day.

History

1678

— Huygens' wavelets: propagation as re-emission, but without phases

1818

— Fresnel's memoir: quadratic-phase zone sums make diffraction computable — the kernel's mathematics, born in optics

1923

— de Broglie assigns matter its wavelength; the dictionary becomes conceivable

1926

— Schrödinger's equation; its free Green function is Fresnel's integral with m/ħt for k/z

1927

— Davisson–Germer and G.P. Thomson diffract electrons — matter obeys the optics

1999

— Arndt's group interferes C₆₀ molecules: kernel spreading and diffraction at 720 amu

Going further

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.6 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 5 · Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Problem 2.22 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Complement G_I · Feynman & Hibbs, Quantum Mechanics and Path Integrals, Ch. 3 · Saleh & Teich, Fundamentals of Photonics, Ch. 4 (Fresnel propagation). The kernel and spreading law are standard results. The Gaussian integral is evaluated with $\alpha=it/2m\hbar$ (using $\sqrt{1/i}=e^{-i\pi/4}$ ); the spreading formula is $\Delta x(t)=\Delta x(0)\sqrt{1+(\hbar t/2m\Delta x(0)^2)^2}$ ; and the Heisenberg-picture variance is $a^2+(\hbar t/2ma)^2$ . The regulated momentum integral matches the kernel under Richardson $\varepsilon\to0^+$ extrapolation, with the $e^{-i\pi/4}$ branch exact; the TDSE residual is identically zero (symbolic); $\int K_0dx'=1$ by regulated quadrature; composition holds to $10^{-12}$ ; the spreading law is confirmed to $10^{-6}$ over three epochs; the Van Vleck identities and the Fresnel dictionary hold exactly (symbolic); and the anchors (43 as/75 as electron; 315 ns/166 nm helium; 22.7 ns/0.26 mm C₆₀) are computed from CODATA masses. The group/phase-velocity relation $v_g=2v_\phi=\hbar k/m$ is treated in Group Velocity and Dispersion.

Builds on: The Quantum Propagator · Momentum Representation · Continuous Spectra and Dirac Delta Normalization · Wave Packet Spreading and Dispersion

Leads to: Propagator for the Harmonic Oscillator · Green Functions in Quantum Mechanics · Semiclassical/WKB methods (APRX) · Matter-wave optics applications

Why doesn't the momentum integral converge without the regulator — and is the answer trustworthy anyway?

The integrand has modulus 1 everywhere (pure phases), so the integral is conditionally convergent at best — Riemann's definition gives it meaning only via a limiting procedure, and the $e^{-\varepsilon p^2}$ regulator is the cleanest one. Trustworthy on three independent grounds: the regulated answer solves the TDSE exactly and satisfies the δ-limit and composition (the certification trio); the same result follows from evolving normalizable packets, where no regulator is needed (the packet's own Gaussian supplies convergence); and physically, $\varepsilon>0$ is an infinitesimal Wick rotation $t\to t-i\varepsilon'$ , damping the unphysical infinite-momentum tail. The $e^{-i\pi/4}$ branch is the visible residue of the choice — and it is measurable, being the ancestor of the Maslov phases of the next article.

The kernel spreads forever — doesn't that violate momentum conservation?

It expresses it. Momentum's distribution is exactly frozen for free flight ( $[\hat{p},\hat{H}]=0$ — the constants-of-the-motion theorem), and because the momentum spread $\sigma_p$ is frozen, positions fan out linearly forever at $\sigma_p/m$ . Spreading is the position-space shadow of conserved momentum dispersion. The one thing that never changes is the packet's momentum histogram — which is why time-of-flight images taken at $t\gg\tau$ measure it faithfully.

Does the wavefunction's phase velocity being "wrong" (v_φ = v_g/2) affect the kernel?

No observable is touched, and the kernel is a good place to see why. The factor-two mismatch between phase and group velocity (owned by the wave chapter's Group Velocity and Dispersion) lives in single plane-wave components; the kernel is an integral over all of them, and every measurement involves the assembled packet, whose envelope moves at the group velocity. In stationary-phase language: the kernel's chirp is dominated, at arrival point $x$ , by the momentum class $p=m(x-x')/t$ — the classical (group) velocity — while phase velocities of individual components are integrated over and never surface alone.

Is there a spreading law for non-Gaussian packets?

The variance version is universal: for any free packet, $\mathrm{Var}\,x(t)=\mathrm{Var}\,x(0)+\frac{2t}{m}\mathrm{Cov}(x,p)_0+\frac{t^2}{m^2}\mathrm{Var}\,p(0)$ — exact, from the Heisenberg solution $\hat{x}_H=\hat{x}+\hat{p}t/m$ . What is special to Gaussians is shape preservation: Gaussian in, Gaussian out (the kernel is a Gaussian convolution). Non-Gaussian packets change shape while their variance follows the same quadratic law; note also the covariance term allows transient narrowing if position and momentum start anti-correlated (a "focused" matter wave) — the atom-optics analogue of a converging beam, and precisely how time-of-flight lenses are engineered.

Where exactly does the classical particle hide inside K₀?

In the stationary phase. Ask which launch momentum dominates the arrival amplitude at $(x,t)$ : the phase $pd/\hbar-p^2t/2m\hbar$ is stationary at $p=md/t$ — precisely the classical momentum for that trip, and the kernel's exponent evaluated there is the classical action. For quadratic Hamiltonians the stationary-phase evaluation is exact (this article's Van Vleck miracle); for general potentials it is the leading asymptotic — the WKB/semiclassical program of the APRX chapter. A classical particle is a stationary-phase point that the rest of the integral has agreed to cancel around.

How does the 3D kernel differ — and does spreading change in higher dimensions?

Cartesian directions factorize for free flight: $K_0^{(3)}=\prod_{i=1}^{3}K_0(x_i,t;x_i')=\left(\frac{m}{2\pi i\hbar t}\right)^{3/2}e^{im|\mathbf{r}-\mathbf{r}'|^2/2\hbar t}$ — same chirp, dimension-counting prefactor (matching the $\delta^3$ initial condition, per the dimension audit habit). Each direction spreads by its own $\sigma_i(t)$ law with its own $\tau_i$ ; an anisotropic trap release therefore inverts its aspect ratio in time of flight (tight direction spreads fastest) — the classic experimental signature by which condensates are distinguished from thermal clouds.