Wave-Packet Time Evolution Notebook

This notebook guide specifies a reproducible calculation for free Gaussian wave-packet time evolution. The goal is not just to make a plot of a spreading packet. The notebook should verify the analytic free-particle formulas, expose the numerical assumptions behind a Fourier-grid method, and record convergence checks that make the output trustworthy.

The existing source notebook notebooks/wave-mechanics-canonical-systems/free-motion/gaussian-wave-packet-spreading.ipynb already implements the core validation target. A Dynamics-specific notebook should keep the same validation discipline while adding the formulation-level interpretation: momentum-space evolution, propagator comparison, expectation values, and uncertainty checks.

Purpose

The notebook should demonstrate four facts:

• A free Gaussian packet remains Gaussian under the free-particle Hamiltonian.
• The packet center moves with the group velocity $p_0/m$.
• The spatial width grows according to the analytic spreading formula.
• A spectral Fourier method reproduces these results when the grid and time window are chosen correctly.

It should also show what can go wrong: periodic wraparound, under-resolved momentum support, inconsistent Fourier conventions, and plots that look plausible while failing moment checks.

Physics Goal

Use the one-dimensional free Hamiltonian

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

The initial state is the normalized Gaussian

$$\psi(x,0) = \left( \frac{1}{2\pi\sigma_x^2} \right)^{1/4} \exp\left[ - \frac{(x-x_0)^2}{4\sigma_x^2} + \frac{ip_0(x-x_0)}{\hbar} \right].$$

The analytic benchmark is

$$\langle x\rangle(t) = x_0+\frac{p_0}{m}t,$$

and

$$\sigma_x(t) = \sigma_x \sqrt{ 1+ \left( \frac{\hbar t}{2m\sigma_x^2} \right)^2 }.$$

The momentum width is constant:

$$\sigma_p(t)=\sigma_p(0) = \frac{\hbar}{2\sigma_x}.$$

Therefore the uncertainty product grows as

$$\sigma_x(t)\sigma_p(t) = \frac{\hbar}{2} \sqrt{ 1+ \left( \frac{\hbar t}{2m\sigma_x^2} \right)^2 }.$$

This is a useful reminder: free spreading is not a loss of norm or a force effect. It is dispersion from the quadratic momentum dependence of the energy.

Mathematical Model

Momentum-space evolution is diagonal:

$$\phi(p,t) = e^{-ip^2t/(2m\hbar)}\phi(p,0).$$

The position-space state is reconstructed by the inverse Fourier transform:

$$\psi(x,t) = \frac{1}{\sqrt{2\pi\hbar}} \int dp\, e^{ipx/\hbar} \phi(p,t).$$

The notebook should state its Fourier convention explicitly and keep the same convention in every analytic formula, grid transform, and validation cell.

For dimensionless benchmark cells, it is reasonable to set

$$\hbar=m=1.$$

Then $p=k$ if the grid variable is the Fourier wavenumber. If the notebook uses physical units, every plotted axis and formula should display the conversion.

Numerical Method

Use a periodic spatial box of length $L$ with $N$ equally spaced grid points:

$$x_j = \left( j-\frac{N}{2} \right)\Delta x, \qquad \Delta x=\frac{L}{N}.$$

The Fourier wavenumbers are

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

ordered according to the FFT convention used by the software library. With $\hbar=1$, the free evolution factor is

$$e^{-ik_n^2t/(2m)}.$$

The recommended algorithm is:

1. Build $\psi(x_j,0)$ on the grid.
2. Normalize using the discrete inner product $\sum_j\lvert\psi_j\rvert^2\Delta x$.
3. FFT to momentum space.
4. Multiply each mode by the exact free phase.
5. Inverse FFT to reconstruct $\psi(x_j,t)$.
6. Measure norm, mean position, variance, and optionally momentum moments.

Because the free spectral step is exact for the represented Fourier modes, there is no time-step error in this basic notebook. The dominant numerical errors come from finite box size, finite grid spacing, Fourier normalization, and periodic boundary effects.

How to Run the Notebook

Use a lightweight environment with NumPy. Plotting libraries are optional for visualization, but validation cells should run without requiring plotting.

Recommended baseline parameters in dimensionless units are:

$$\hbar=1, \qquad m=1, \qquad \sigma_x=2, \qquad x_0=-20, \qquad k_0=1.5.$$

Choose a box large enough that the packet remains far from the periodic boundary over the time interval being tested. A typical first run is

$$N=2048, \qquad L=200.$$

Run the notebook top to bottom and record:

• Python and NumPy versions;
• $N$, $L$, $\Delta x$, and $\Delta k$;
• packet parameters $x_0$, $\sigma_x$, and $p_0$ or $k_0$;
• sampled times;
• validation tolerances;
• whether plots were generated or only numerical checks were run.

Expected Results

The norm should remain close to one:

$$\left\lvert 1-\sum_j\lvert\psi_j(t)\rvert^2\Delta x \right\rvert \lt\epsilon_{\rm norm}.$$

The mean position should follow

$$\langle x\rangle_{\rm num}(t) \approx x_0+\frac{p_0}{m}t.$$

The numerical width should follow

$$\sigma_{x,\rm num}(t) \approx \sigma_x(t).$$

For a well-resolved grid, the relative width error should decrease as the box and grid are refined, until floating-point or boundary effects dominate.

Expected plots include:

• $\lvert\psi(x,t)\rvert^2$ at several times;
• packet center versus time compared with $x_0+p_0t/m$;
• width versus time compared with the analytic formula;
• optional momentum density showing that $\lvert\phi(p,t)\rvert^2$ is time independent;
• convergence of norm and width error as $N$ and $L$ are varied.

Validation Checks

The minimum validation suite should include:

Check	Target
Norm conservation	$\sum_j\lvert\psi_j(t)\rvert^2\Delta x=1$ within tolerance
Group velocity	$\langle x\rangle(t)-x_0=p_0t/m$
Variance growth	$\sigma_x(t)$ matches the analytic Gaussian formula
Momentum conservation	$\langle p\rangle(t)=p_0$ and $\sigma_p(t)=\hbar/(2\sigma_x)$
Fourier-grid convergence	errors shrink under sensible increases of $N$ and $L$

Do not count a plot as validation. The notebook should use assertions or printed residuals with stated tolerances.

Convergence Checks

Vary $N$ at fixed physical box size to test grid spacing:

$$\Delta x=\frac{L}{N}.$$

Vary $L$ at fixed resolution strategy to test boundary effects. A larger box should not change the early-time moments if the packet remains far from the boundary.

Check that the momentum distribution is well inside the represented range. The effective maximum wavenumber is the Nyquist scale:

$$k_{\rm max}\sim\frac{\pi}{\Delta x}.$$

The packet’s momentum support around $k_0$ should be small compared with this cutoff. If high-$k$ tails are under-resolved, the packet can appear to propagate with the wrong width or develop spurious oscillations.

Common Numerical Failures

• Letting the packet hit the periodic boundary and mistaking wraparound for physical interference.
• Using a Fourier convention inconsistent with the analytic momentum formula.
• Forgetting the factor $\Delta x$ in discrete norm and moment calculations.
• Choosing $L$ large but $N$ too small, producing poor momentum resolution.
• Choosing $N$ large but $L$ too small, producing boundary contamination.
• Validating only norm conservation while ignoring mean and variance.
• Centering the packet too close to the boundary for the chosen time interval.

Extensions

Natural extensions include:

• compare Fourier evolution with convolution using the Free-Particle Propagator;
• compute the Wigner function of the evolving packet and verify phase-space shear;
• add a chirped Gaussian with nonzero initial position-momentum covariance;
• compare a Gaussian packet with a non-Gaussian packet whose width does not have the same analytic formula;
• introduce a split-operator method for a weak potential and separate spectral error from time-step error.

Cross-Links

• Gaussian Wave Packets
• Wave Packet Spreading
• Free-Particle Propagator
• Free-Particle Propagator Notebook
• Wigner Function
• Fast Fourier Transform
• How to Use Computational Notebooks
• Validation Tests
• Computational Environments

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.

Exercises

1. Why is norm conservation not enough to validate this notebook?

Solution

A Fourier method with a unit-modulus phase factor can conserve norm even if the grid is too small, the Fourier convention is inconsistent, or the packet has wrapped around the periodic boundary. The notebook must also check the group velocity, width growth, and convergence with respect to $N$ and $L$.

2. In dimensionless units with $\hbar=m=1$, $\sigma_x=2$, and $p_0=1.5$, what are the expected center and width at $t=10$ if $x_0=-20$?

Solution

The center is

$$\langle x\rangle(10) = -20+1.5(10) = -5.$$

The width is

$$\sigma_x(10) = 2\sqrt{ 1+ \left( \frac{10}{2(2)^2} \right)^2 } = 2\sqrt{1+\left(\frac{10}{8}\right)^2}.$$

Thus

$$\sigma_x(10) = 2\sqrt{2.5625} = 3.20156\ldots.$$

3. What happens if the packet reaches the periodic boundary of the FFT box?

Solution

The FFT grid represents a periodic domain. If the packet reaches the boundary, it reenters from the other side. That wraparound is a numerical artifact for an infinite-line free-particle benchmark. It can change the computed moments and create interference that is not part of the intended physical problem.

4. Why does the momentum probability density stay fixed while the position density spreads?

Solution

For a free particle,

$$\phi(p,t) = e^{-ip^2t/(2m\hbar)}\phi(p,0).$$

The phase has unit magnitude, so

$$\lvert\phi(p,t)\rvert^2 = \lvert\phi(p,0)\rvert^2.$$

The position density changes because different momentum components acquire different phases, and their interference pattern in position space changes with time.