Gaussian Wave Packets, Propagators, and the Semiclassical Limit

The free Schr"odinger equation is one of the cleanest places to study the geometry of quantum evolution. The evolving object is a state in Hilbert space, and the familiar picture of motion has to be reconstructed from that state.

i\hbar\,\partial_t \psi(x,t)= -\frac{\hbar^2}{2m}\partial_x^2 \psi(x,t)

What matters here is how several descriptions of the same evolution fit together:

spectral decomposition in momentum space
integral-kernel propagation in position space
asymptotic reconstruction through stationary phase
semiclassical concentration around classical rays

The Gaussian packet is useful because it keeps those viewpoints connected while staying exactly solvable.

A state in Hilbert space and its projection into the position representation.

The packet as a state in Hilbert space

Take an initial wave packet in one spatial dimension:

\psi(x,0)=\frac{1}{(2\pi\sigma_0^2)^{1/4}}\exp\!\left[-\frac{(x-x_0)^2}{4\sigma_0^2}+ik_0x\right]

This state is square integrable, sharply localized in both position and momentum, and it saturates the Heisenberg bound at the initial time:

\Delta x(0)\,\Delta p(0)=\frac{\hbar}{2}

There are two reasons this state matters.

First, its Fourier transform is also Gaussian, so the momentum-space description stays transparent:

\widetilde\psi(k,0)=\left(\frac{2\sigma_0^2}{\pi}\right)^{1/4}\exp\!\left[-\sigma_0^2(k-k_0)^2-i(k-k_0)x_0\right]

Second, the free Hamiltonian is diagonal in momentum space:

H=\frac{\hat p^2}{2m},\qquad E(k)=\frac{\hbar^2k^2}{2m}

Time evolution is therefore a phase rotation for each mode:

\widetilde\psi(k,t)=e^{-iE(k)t/\hbar}\widetilde\psi(k,0)

The entire phenomenon of spreading comes from the fact that this phase is quadratic in $k$ . Different momentum components rotate at different rates, so a packet that begins concentrated in position space eventually broadens.

Exact free propagation

Returning to position space gives

\psi(x,t)=\int_{-\infty}^{\infty}\frac{dk}{\sqrt{2\pi}}\,e^{ikx-i\hbar k^2 t/(2m)}\widetilde\psi(k,0)

For the Gaussian initial state, the integral is elementary and yields another Gaussian:

\psi(x,t)=\frac{1}{(2\pi\sigma_0^2)^{1/4}}\frac{1}{\sqrt{1+i\hbar t/(2m\sigma_0^2)}}\exp\!\left[ik_0x-\frac{i\hbar k_0^2 t}{2m}-\frac{\left(x-x_0-\hbar k_0 t/m\right)^2}{4\sigma_0^2\left(1+i\hbar t/(2m\sigma_0^2)\right)}\right]

The center of the packet moves with the group velocity

v_g=\frac{1}{\hbar}\frac{dE}{dk}=\frac{\hbar k_0}{m}

while the width evolves as

\Delta x(t)=\sigma_0\sqrt{1+\frac{\hbar^2 t^2}{4m^2\sigma_0^4}}

That square root is the exact statement of dispersive broadening. It also shows how localization and momentum spread are tied together. A tightly squeezed initial state carries a broad momentum distribution, and the Hamiltonian turns that distribution into phase winding.

Gaussian Wave Packet

Free-particle spreading with units chosen so m = \hbar = 1.

Initial width 1.20Central momentum 2.40Time 0.80

Packet width

1.245

Uncertainty product

0.519

The propagator viewpoint

The same evolution can be written through the kernel of the unitary group:

\psi(x_f,t)=\int_{-\infty}^{\infty} dx_i\,K(x_f,t;x_i,0)\psi(x_i,0)

For the free particle,

K(x_f,t;x_i,0)=\sqrt{\frac{m}{2\pi i\hbar t}}\exp\!\left[\frac{im(x_f-x_i)^2}{2\hbar t}\right]

This kernel already contains the semiclassical structure. Its phase is the classical action for a straight path from $x_i$ to $x_f$ in time $t$ :

S_{\mathrm{cl}}[x]=\frac{m(x_f-x_i)^2}{2t}

Even before path integrals enter the discussion, the free kernel already points to the general rule:

K\sim A\,e^{iS_{\mathrm{cl}}/\hbar}

The prefactor is not decorative. It measures how nearby trajectories focus or defocus and becomes the one-dimensional version of the Van Vleck determinant.

The propagator sums amplitudes over all paths, not probabilities over a few selected trajectories.

Stationary phase and the semiclassical limit

The semiclassical regime is an asymptotic reading of the exact theory. Write the momentum integral schematically as

\psi(x,t)=\int dk\,a(k)\,e^{\frac{i}{\hbar}\Phi(k;x,t)}

with phase

\Phi(k;x,t)=\hbar kx-\frac{\hbar^2k^2}{2m}t

When $\hbar$ is small compared with the action scales in the problem, the integral is dominated by stationary points:

\frac{d\Phi}{dk}=0\quad\Longrightarrow\quad x=\frac{\hbar k}{m}t

This is precisely the classical ray condition. The packet amplitude is largest near the locus that classical mechanics would assign to momentum $p=\hbar k$ . Stationary phase therefore does not say that quantum evolution becomes classical everywhere. It says that away from critical points, destructive interference suppresses contributions, while near critical points the phases line up long enough to leave macroscopic weight.

The first correction comes from the Hessian:

\psi(x,t)\approx a(k_\star)e^{\frac{i}{\hbar}\Phi(k_\star)}\sqrt{\frac{2\pi\hbar}{i\Phi''(k_\star)}}

For the free problem this is exact enough to recover the dispersive scaling. In more structured systems, the same machinery becomes the bridge between spectral theory, WKB analysis, and microlocal geometry.

Why Gaussian packets remain special

It is tempting to treat the Gaussian as a pedagogical convenience. That misses how structurally useful it is.

The Gaussian is the unique state that simultaneously:

saturates the uncertainty inequality at the initial time
remains Gaussian under free evolution and harmonic confinement
makes the symplectic geometry of phase-space transport almost explicit

In the language of coherent states, the packet is the closest quantum analogue to a classical phase-space point while still respecting the noncommutativity of $\hat x$ and $\hat p$ .

This is why Gaussian packets reappear everywhere: coherent-state path integrals, paraxial optics, semiclassical scattering, quantum information, and linearized quantum field theory around stable vacua.

A field-theoretic remark

The same structure survives in relativistic settings after suitable reinterpretation. For a free scalar field, mode operators evolve independently:

\phi(x)=\int\!\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\mathbf p}}}\left(a_{\mathbf p}e^{-ip\cdot x}+a_{\mathbf p}^\dagger e^{ip\cdot x}\right)

Wave packets then appear as smeared one-particle states,

|f\rangle=\int\!\frac{d^3p}{(2\pi)^3}\,f(\mathbf p)\,a_{\mathbf p}^\dagger|0\rangle

and the same competition between localization and spectral width returns. The only difference is that the dispersion relation is now relativistic:

E_{\mathbf p}=\sqrt{\mathbf p^2c^2+m^2c^4}

The packet no longer spreads according to a simple quadratic phase, yet the asymptotic reasoning still runs through stationary points of the phase.

What this buys you in practice

A lot of advanced quantum theory becomes easier once you internalize this one example properly.

Scattering theory becomes more intuitive because incoming states are packets, not plane waves.
Semiclassical approximations become less mysterious because the stationary points already encode classical trajectories.
Propagators stop feeling abstract once you can read their phase as action and their prefactor as transport data.

The Gaussian packet is a compact laboratory for unitary evolution, asymptotics, and the quantum-to-classical boundary. It is simple enough to solve exactly and rich enough to expose the geometry underneath the formalism.

A bright field beneath a dark sky with the Earth hanging in the distance — Quantum theory gets stranger the closer you look.