Free-Particle Propagator: First Encounter

The free-particle propagator evolves a wavefunction by adding amplitudes from every initial position. This first example connects Fourier evolution to motion in coordinate space; Propagator Kernel develops the general derivation and composition law.

For a particle on the line with $H=p^2/(2m)$ and $t>0$,

$$K_0(x,t;x',0) = \left( \frac{m}{2\pi i\hbar t} \right)^{1/2} \exp\left[ \frac{im(x-x')^2}{2\hbar t} \right].$$

It evolves an initial wavefunction by

$$\psi(x,t) = \int_{-\infty}^{\infty} K_0(x,t;x',0)\psi(x',0)\,dx'.$$

The kernel is a transition amplitude, not a probability density. Its units in one dimension are inverse length, so the integral has the same units as the wavefunction.

For $t>0$, choose $1/\sqrt{i}=e^{-i\pi/4}$. This phase is necessary to recover the initial state in the limit $t\to0^+$.

Helpful background. Free Particle to use the plane-wave dispersion relation.

Free propagation and classical action

The equal-time limit is distributional:

$$K_0(x,0;x',0)=\delta(x-x').$$

For nonzero $t$, the kernel is oscillatory rather than localized. The exponent is the classical free-particle action,

$$S_{\mathrm{cl}} = \frac{m(x-x')^2}{2t}, \qquad K_0\propto e^{iS_{\mathrm{cl}}/\hbar}.$$

This phase is the first bridge to stationary phase and path integrals. It does not mean that the quantum particle follows one selected classical trajectory: the final amplitude integrates contributions from every initial coordinate.

Wave-Packet Use

Applying $K_0$ to a Gaussian packet reproduces its exact free spreading. A narrow packet contains a broad momentum distribution, whose components accumulate different phases under $E=p^2/(2m)$. The kernel packages that Fourier evolution into one coordinate-space convolution.

Use Gaussian Wave Packets for the packet calculation and Wave-Packet Spreading for its interpretation. Use the general propagator treatment for the derivation, branch prescription, composition law, higher-dimensional kernels, boundary conditions, and path-integral connection.

Immediate Cautions

• Keep the normalization factor and its real-time square-root phase.
• Integrate over the initial coordinate; $K_0$ alone is not the evolved state.
• Do not use the free-space kernel unchanged when boundaries or potentials are present.
• Interpret the $t\to0$ limit as a delta distribution, not pointwise convergence.

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw–Hill, 1965, Chapter 2.