Free-Particle Propagator Notebook

This notebook guide turns the exact free-particle propagator into a numerical consistency test. It evolves one initial Gaussian by three routes: the analytic Gaussian formula, direct quadrature of the full-line kernel, and diagonal evolution on a periodic Fourier grid. Agreement among all three is much stronger evidence than a plausible-looking animation.

The derivation of the kernel belongs to Free-Particle Propagator, and the physics of packet spreading belongs to Gaussian Wave Packets and Wave-Packet Spreading. This page focuses on implementation, validation, and the boundary-condition distinction between the infinite line and a finite FFT box.

Purpose

The notebook should establish that:

• the full-line propagator convolves an initial wavefunction into the correct evolved state;
• direct kernel quadrature and Fourier-space evolution agree when they approximate the same physical problem;
• an FFT implements a periodic-box propagator, not a literal finite sample of the full-line kernel;
• the kernel’s complex phase carries essential information even though its magnitude is spatially constant;
• convergence requires separate tests of grid spacing, box size, oscillatory phase resolution, and boundary contamination.

The notebook should preserve complex amplitudes until the final observable is formed. Comparing only probability densities can conceal a wrong square-root branch or a global phase error.

Physics Goal

For the one-dimensional Hamiltonian

$$H=\frac{p^2}{2m},$$

the full-line kernel for $t\gt0$ is

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

The branch continuous with unitary forward evolution is

$$\left( \frac{1}{i} \right)^{1/2} = e^{-i\pi/4}.$$

The propagated state is

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

The notebook should not infer this formula from its numerical output. It should take the exact kernel as an analytic input and test whether each discretization realizes its action correctly.

Analytic Gaussian Benchmark

Use the normalized initial packet

$$\psi_0(x) = \frac{1}{(2\pi\sigma_0^2)^{1/4}} \exp\left[ - \frac{(x-x_0)^2}{4\sigma_0^2} + \frac{ip_0(x-x_0)}{\hbar} \right].$$

Define

$$\tau = \frac{\hbar t}{2m\sigma_0^2}, \qquad x_c(t) = x_0+\frac{p_0t}{m}.$$

The exact evolved wavefunction is

$$\begin{aligned} \psi_{\rm G}(x,t) &= \frac{1}{(2\pi\sigma_0^2)^{1/4}} \frac{1}{\sqrt{1+i\tau}} \\ &\quad\times \exp\left[ - \frac{(x-x_c(t))^2} {4\sigma_0^2(1+i\tau)} + \frac{i}{\hbar} \left( p_0(x-x_0)-\frac{p_0^2t}{2m} \right) \right]. \end{aligned}$$

Choose the square root continuously from $\sqrt{1+i\tau}=1$ at $t=0$. Its probability density is Gaussian with

$$\langle x\rangle(t)=x_c(t), \qquad \sigma_x(t) = \sigma_0\sqrt{1+\tau^2}.$$

This benchmark tests amplitude, phase, translation, and spreading simultaneously. The Wave-Packet Time Evolution Notebook develops the packet observables in more detail; here they serve as independent checks on the kernel calculation.

Three Evolution Routes

All routes must begin from the same sampled initial state and be compared on the same position grid.

Analytic route

Evaluate $\psi_{\rm G}(x_j,t)$ directly from the exact expression. Do not renormalize it after sampling. A sampled norm different from one is useful diagnostic information about finite box size and grid resolution.

Full-line kernel quadrature

On a uniform grid with spacing $\Delta x$, approximate the convolution by

$$\psi_{{\rm dir},j}(t) \approx \Delta x \sum_{\ell=0}^{N-1} K_\infty(x_j,t;x_\ell,0) \psi_0(x_\ell).$$

This route has two distinct approximations:

1. the integral over the infinite line is truncated to the finite computational interval;
2. the resulting oscillatory integral is replaced by a quadrature sum.

The dense operation costs $O(N^2)$ work and $O(N^2)$ memory if the entire kernel matrix is stored. Evaluate rows in blocks when memory is limited. Blocking changes neither the mathematical approximation nor the expected answer.

Do not use $t=0$ in this formula. The kernel tends to a delta distribution, not an ordinary function, and its increasingly rapid oscillations make naive quadrature least reliable precisely when $t$ is small.

Periodic Fourier route

Take a periodic interval of length $L$ with

$$x_j=x_{\min}+j\Delta x, \qquad \Delta x=\frac{L}{N}, \qquad j=0,\ldots,N-1.$$

There is no duplicate endpoint. The represented wavenumbers are

$$k_n=\frac{2\pi n}{L},$$

ordered according to the FFT library. The free evolution of each mode is

$$\widetilde\psi_n(t) = \exp\left[ - \frac{i\hbar k_n^2t}{2m} \right] \widetilde\psi_n(0).$$

The algorithm is therefore:

1. normalize $\psi_0(x_j)$ with the weighted grid norm;
2. transform it with the forward FFT;
3. multiply every represented mode by the exact free phase;
4. apply the inverse FFT;
5. retain the complex result without post-evolution renormalization.

For a time-independent free particle, this is one exact spectral step for the represented periodic modes. There is no time-step error. Disagreement comes from finite spatial bandwidth, periodic boundaries, transform conventions, or an implementation mistake.

Infinite-Line and Periodic Kernels

The distinction between the two kernels is the conceptual center of the notebook. On a circle of circumference $L$, the exact periodic kernel is

$$K_L(x,t;x',0) = \frac{1}{L} \sum_{n\in\mathbb Z} \exp\left[ ik_n(x-x') - \frac{i\hbar k_n^2t}{2m} \right].$$

Formally, it can also be written as an image sum,

$$K_L(x,t;x',0) = \sum_{r\in\mathbb Z} K_\infty(x-x'+rL,t;0,0),$$

with the same real-time convergence prescription as the full-line kernel. The image terms are the amplitudes for propagation around the periodic domain by different windings.

Keeping only the $N$ Fourier modes represented on the grid gives

$$K_{L,N}(x_j,t;x_\ell,0) = \frac{1}{L} \sum_{n\in\mathcal I_N} \exp\left[ ik_n(x_j-x_\ell) - \frac{i\hbar k_n^2t}{2m} \right],$$

where $\mathcal I_N$ is the FFT mode index set. Its grid convolution is

$$\psi_j(t) = \Delta x \sum_{\ell=0}^{N-1} K_{L,N}(x_j,t;x_\ell,0) \psi_\ell(0).$$

Because $\Delta x/L=1/N$, this is exactly the discrete Fourier algorithm, up to the stated transform normalization. The matrix

$$U_{j\ell}(t) = \Delta x\,K_{L,N}(x_j,t;x_\ell,0)$$

is unitary on the discrete grid when every represented mode is retained. By contrast, the matrix formed by sampling $K_\infty$ on a finite interval is generally not exactly unitary.

The two routes agree before periodic images matter if the initial and evolved packet are negligible near the boundaries and the full-line quadrature is resolved. They are not expected to agree after the packet wraps around the box.

Grid and Baseline Parameters

A useful dimensionless baseline is

$$\hbar=m=1, \qquad \sigma_0=2, \qquad x_0=-15, \qquad p_0=1.$$

Start with

$$L=80, \qquad N=2048, \qquad t=8.$$

Then the analytic center is $x_c=-7$, the width is $\sigma_x=2\sqrt2$, and the packet remains well separated from either boundary. This time is also large enough that the full-line kernel is not excessively oscillatory at the baseline grid spacing. The values are a starting point, not a substitute for convergence tests.

Record in the notebook:

• $\hbar$, $m$, $x_0$, $p_0$, and $\sigma_0$;
• $x_{\min}$, $L$, $N$, $\Delta x$, and the FFT wavenumber ordering;
• every evaluation time;
• the floating-point dtype and software versions;
• the source and output blocks used for direct convolution;
• all validation tolerances and the norm used to define each error.

Notebook Workflow

Organize the calculation into reproducible cells:

1. define parameters and construct a nonduplicated periodic grid;
2. build and discretely normalize the initial Gaussian;
3. verify that its boundary probability is negligible;
4. evaluate the analytic Gaussian benchmark;
5. evaluate the full-line kernel convolution in memory-safe blocks;
6. evolve the same array by one FFT spectral step;
7. compute complex wavefunction errors, density errors, norms, moments, and overlaps;
8. visualize the kernel magnitude and phase;
9. repeat for grid and box refinements;
10. run assertions before producing presentation plots.

Keep computation and plotting separate. A validation cell should still fail clearly when plots are disabled.

Complex Kernel Phase

For $t\gt0$, write the full-line kernel as

$$K_\infty = \left( \frac{m}{2\pi\hbar t} \right)^{1/2} e^{i\phi_\infty},$$

where

$$\phi_\infty(x;x',t) = \frac{m(x-x')^2}{2\hbar t} - \frac{\pi}{4}.$$

Its magnitude,

$$\lvert K_\infty\rvert = \left( \frac{m}{2\pi\hbar t} \right)^{1/2},$$

is independent of separation. Localization emerges from destructive and constructive interference in the convolution, not from a decaying kernel magnitude.

For a fixed source point $x'$, include four plots:

• the constant magnitude $\lvert K_\infty\rvert$;
• the real and imaginary parts;
• the wrapped phase $\operatorname{Arg}K_\infty$ in a cyclic color scale or on $(-\pi,\pi]$;
• the analytic unwrapped quadratic phase $\phi_\infty$.

The wrapped phase contains discontinuous jumps by $2\pi$ that are not physical singularities. Do not apply a generic phase-unwrapping routine and assume it reconstructed the branch correctly; compare with the known analytic phase. Plot the periodic kernel separately if desired, because interference among images makes both its magnitude and phase nontrivial.

Phase-Resolution Criterion

The phase change between neighboring source-grid points is approximately

$$\Delta\phi \approx \left\lvert \frac{\partial\phi_\infty}{\partial x'} \right\rvert \Delta x = \frac{m\lvert x-x'\rvert}{\hbar t} \Delta x.$$

For all source and output points used in a comparison, require this change to be comfortably smaller than $\pi$. A conservative study should demonstrate convergence as its maximum value decreases, rather than declaring one fixed threshold universally sufficient.

This estimate explains a counterintuitive feature of direct real-time quadrature: decreasing $t$ makes the kernel harder to sample. The FFT route does not sample this coordinate-space oscillation directly; it represents the exact phase of each retained momentum mode instead.

Validation Metrics

Use the discrete inner product

$$\langle f|g\rangle_{\Delta x} = \Delta x\sum_j f_j^*g_j$$

and norm

$$\lVert f\rVert_{\Delta x} = \sqrt{\langle f|f\rangle_{\Delta x}}.$$

For two routes $a$ and $b$, define the phase-sensitive error

$$\epsilon_{a,b} = \lVert\psi_a-\psi_b\rVert_{\Delta x}.$$

Also report the normalized fidelity

$$F_{a,b} = \frac{ \lvert\langle\psi_a|\psi_b\rangle_{\Delta x}\rvert^2 }{ \lVert\psi_a\rVert_{\Delta x}^2 \lVert\psi_b\rVert_{\Delta x}^2 }.$$

Fidelity is insensitive to a global phase, whereas $\epsilon_{a,b}$ is not. Both are needed when testing the complex prefactor of a propagator.

The minimum validation table is:

Check	Diagnostic	Expected behavior
Initial sampling	$\lVert\psi_0\rVert_{\Delta x}$ and boundary mass	near one before optional discrete normalization; boundary mass converges to zero
FFT unitarity	$\lVert\psi_{\rm FFT}(t)\rVert_{\Delta x}-1$	near floating-point roundoff
Analytic agreement	$\epsilon_{{\rm FFT},{\rm G}}$	decreases under grid and box refinement
Direct agreement	$\epsilon_{{\rm dir},{\rm G}}$	decreases when truncation and phase resolution improve
Route agreement	$\epsilon_{{\rm dir},{\rm FFT}}$	converges before periodic images contribute
Center	$\langle x\rangle-x_c(t)$	approaches zero
Width	$\sigma_{x,\rm num}-\sigma_x(t)$	approaches zero
Composition	two successive evolutions versus one combined evolution	near roundoff for the FFT route
Branch phase	complex error and analytic kernel phase	catches errors hidden by density plots

For the composition test, choose positive $t_1$ and $t_2$ and compare

$$U(t_2)U(t_1)\psi_0$$

with

$$U(t_1+t_2)\psi_0.$$

The periodic spectral route should satisfy this identity to roundoff because its mode phases multiply exactly. The direct quadrature route tests the finite-domain approximation to the continuum Composition Law and generally converges more slowly.

Convergence Study

Vary one numerical scale at a time:

1. Grid refinement at fixed box. Use $N=512,1024,2048$, and increase further if the direct phase remains under-resolved.
2. Box enlargement at fixed spacing. Increase $L$ while keeping $\Delta x$ approximately fixed. This isolates truncation and periodic-image errors from resolution.
3. Time scan. Compare moderate times, a deliberately difficult small time, and a later time approaching wraparound.
4. Source truncation. If direct convolution omits initial points below a magnitude threshold, tighten that threshold and confirm stable results.
5. Block size. Change only the direct-kernel block size. The result should be unchanged to roundoff.

Plot each error against the relevant resolution scale. A single fine-grid result does not establish convergence. If an error stops decreasing, inspect boundary probability, momentum-space tails near the Nyquist mode, phase resolution, and floating-point cancellation before increasing $N$ again.

Expected Results

For a well-resolved isolated packet:

• the analytic, direct, and FFT wavefunctions overlap visually in both real and imaginary parts;
• the complex route errors decrease systematically with refinement;
• the FFT norm and composition residual stay near roundoff;
• the direct-convolution norm approaches one but is not exactly preserved at finite resolution;
• the measured center and width follow the analytic formulas;
• the full-line kernel magnitude is constant while its phase oscillates quadratically;
• the direct route deteriorates first as $t$ becomes small;
• the FFT and full-line routes separate when periodic wraparound becomes appreciable.

That final separation is expected physics for different boundary conditions, not evidence that one formula is intrinsically wrong.

Common Numerical Failures

• Sampling the wrong kernel. A finite FFT box evolves with $K_{L,N}$, not with a rectangular sample of $K_\infty$.
• Duplicating the endpoint. Including both ends of a periodic interval changes the spacing and corrupts the Fourier grid.
• Missing the factor of $2\pi$. FFT frequencies are often returned in cycles per unit length; convert them to angular wavenumbers before using $p=\hbar k$.
• Choosing the wrong square-root branch. Density plots can remain unchanged while the complex wavefunction acquires the wrong global phase.
• Using the coordinate kernel at zero time. Its initial condition is distributional.
• Renormalizing every result. This hides quadrature loss, boundary truncation, and transform-scaling mistakes.
• Comparing only densities. Equal densities do not imply equal states.
• Ignoring periodic wraparound. A packet crossing one boundary re-enters through the other.
• Resolving the packet but not the kernel phase. Smooth initial data do not guarantee accurate direct Fresnel quadrature.
• Materializing an unnecessary dense matrix. Blocked multiplication avoids excessive memory without altering the calculation.
• Changing $L$ and $N$ together without tracking $\Delta x$. The resulting error change cannot be attributed to one cause.

Physical Interpretation

The direct kernel and momentum-space routes are two representations of the same unitary operator on the infinite line. Numerically, however, they expose different approximations. Direct quadrature must resolve cancellation among rapidly varying coordinate-space phases. Fourier propagation diagonalizes the Hamiltonian but replaces the line by a band-limited periodic space.

This is why agreement is most informative in a controlled overlap regime: the packet is far from the boundaries, its momentum support is below the grid cutoff, and the coordinate-space kernel phase is resolved. Outside that regime, the disagreement itself identifies which mathematical idealization has changed.

Extensions

• Replace the Gaussian by a compact smooth packet and compare only the direct and FFT routes.
• Construct $K_{L,N}$ explicitly for a small grid and verify discrete unitarity entry by entry.
• Compare the finite mode sum with a regularized image sum.
• Repeat the calculation in two dimensions using the product kernel.
• Introduce a constant force and compare with the corresponding exact quadratic-action kernel.
• Continue to the Harmonic-Oscillator Propagator, where caustic phases add a new branch-tracking issue.

Cross-Links

• Free-Particle Propagator
• Propagator Kernel
• Composition Law
• Propagators and Boundary Conditions
• Gaussian Wave Packets
• Wave-Packet Spreading
• Fast Fourier Transform
• Numerical Quadrature
• Convergence Tests
• Validation Tests

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective, University Science Books, 2007.
• M. D. Feit, J. A. Fleck Jr., and A. Steiger, “Solution of the Schrödinger Equation by a Spectral Method,” Journal of Computational Physics 47, 412–433 (1982), doi:10.1016/0021-9991(82)90091-2.
• L. N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics, 2000, especially Chapter 3 on periodic grids and the FFT.
• NumPy Developers, Discrete Fourier Transform documentation, consulted for transform normalization and frequency ordering.

Exercises

1. Derive the discrete spectral kernel

Starting from the finite Fourier expansion of a periodic grid state, derive $K_{L,N}$ and show that multiplying it by $\Delta x$ gives the matrix implemented by an FFT, modewise phase multiplication, and an inverse FFT.

Solution

Write the grid state as

$$\psi_j = \frac{1}{N} \sum_{n\in\mathcal I_N} \widetilde\psi_n e^{ik_n(x_j-x_{\min})}.$$

After free evolution,

$$\psi_j(t) = \frac{1}{N} \sum_n e^{ik_n(x_j-x_{\min})} e^{-i\hbar k_n^2t/(2m)} \widetilde\psi_n(0).$$

Insert

$$\widetilde\psi_n(0) = \sum_{\ell=0}^{N-1} \psi_\ell(0) e^{-ik_n(x_\ell-x_{\min})}.$$

Then

$$\psi_j(t) = \sum_\ell \left[ \frac{1}{N} \sum_n e^{ik_n(x_j-x_\ell)} e^{-i\hbar k_n^2t/(2m)} \right] \psi_\ell(0).$$

Since $\Delta x/L=1/N$, the expression in brackets is $\Delta x\,K_{L,N}(x_j,t;x_\ell,0)$. This is exactly the forward-transform, diagonal-phase, inverse-transform algorithm.

2. Estimate direct-quadrature resolution

Let the source and output points satisfy $\lvert x-x'\rvert\le R$. Derive a local phase-resolution estimate and determine how the required $N$ scales as $t\to0^+$ at fixed $L$ and $R$.

Solution

The kernel phase is

$$\phi_\infty = \frac{m(x-x')^2}{2\hbar t} - \frac{\pi}{4}.$$

Therefore

$$\left\lvert \frac{\partial\phi_\infty}{\partial x'} \right\rvert \le \frac{mR}{\hbar t}.$$

With $\Delta x=L/N$,

$$\Delta\phi_{\max} \lesssim \frac{mRL}{\hbar tN}.$$

Requiring $\Delta\phi_{\max}\le\eta$ for some chosen $\eta\ll\pi$ gives

$$N \gtrsim \frac{mRL}{\eta\hbar t}.$$

Thus the direct coordinate-space resolution requirement grows as $1/t$ when $t$ approaches zero.

3. Detect a branch error

Suppose a wrong square-root branch multiplies the correct propagated state by $e^{i\theta}$. For a normalized state, compute the complex wavefunction error and the fidelity relative to the correct state.

Solution

The phase-sensitive error is

$$\begin{aligned} \epsilon &= \lVert e^{i\theta}\psi-\psi\rVert \\ &= \lvert e^{i\theta}-1\rvert \\ &= 2\left\lvert \sin\frac{\theta}{2} \right\rvert. \end{aligned}$$

The normalized fidelity is

$$F = \lvert\langle\psi|e^{i\theta}\psi\rangle\rvert^2 = 1.$$

The density is also unchanged. Fidelity and density checks alone therefore cannot validate the kernel’s complex prefactor.

4. Explain exact discrete composition

Why should the periodic FFT route satisfy composition to roundoff even though it approximates the infinite-line problem only on a finite box?

Solution

Each represented mode evolves by

$$u_n(t) = e^{-i\hbar k_n^2t/(2m)}.$$

The mode factors obey

$$u_n(t_2)u_n(t_1) = u_n(t_1+t_2)$$

exactly in algebra. The FFT and inverse FFT merely change basis, so the finite-dimensional periodic evolution satisfies the group law. Floating-point transforms and phase evaluation introduce only roundoff. This proves composition for the discrete periodic model; it does not remove the difference between that model and the full line.

5. Design a boundary-condition test

Give a numerical experiment that distinguishes loss of direct-quadrature accuracy from genuine periodic wraparound.

Solution

First refine $N$ at fixed $L$. If the direct result approaches the analytic full-line solution while the FFT result remains stable, the original discrepancy was quadrature resolution. Next enlarge $L$ while keeping $\Delta x$ approximately fixed. If the time at which the FFT result departs from the full-line benchmark moves later and the boundary probability decreases, the discrepancy was periodic wraparound. Monitoring the packet probability in fixed boundary strips makes this diagnosis quantitative.