Wigner Function Notebook

This notebook guide computes Wigner functions from wavefunctions and then tests the result more aggressively than a phase-space plot can. A Gaussian state supplies exact normalization, marginals, covariance, and purity benchmarks. Harmonic-oscillator evolution supplies an exact phase-space rotation. A coherent superposition of separated packets supplies interference fringes and genuine negativity.

The definition and interpretation of the Wigner function belong to Wigner Function. Marginals and Quasi-Probabilities owns the meaning of marginals and negativity, while Phase-Space Dynamics owns the Wigner–Moyal evolution equation. This page owns the numerical transform, grid conventions, validation suite, and failure diagnosis.

Purpose

The notebook should demonstrate that:

• a relative-coordinate Fourier transform reconstructs the Wigner function with the intended normalization;
• position and momentum marginals agree with independently computed Born distributions;
• a Gaussian Wigner function is real, normalized, pure, and nonnegative within numerical tolerance;
• harmonic evolution rotates a Wigner function exactly in scaled phase space;
• a coherent superposition produces resolved negative interference fringes;
• relative-coordinate truncation, half-grid interpolation, FFT scaling, and momentum aliasing are independent error sources;
• small negative pixels are not evidence of physical Wigner negativity until they converge under refinement.

Every plotted Wigner function should have a numerical validation record. Color alone cannot establish normalization, purity, or negativity.

Convention

Use the same convention as the canonical phase-space pages:

$$\begin{aligned} W_\psi(x,p) &= \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty}dy\, e^{-ipy/\hbar} \\ &\quad\times \psi\left(x+\frac{y}{2}\right) \psi^*\left(x-\frac{y}{2}\right). \end{aligned}$$

Equivalently, with $s=y/2$,

$$W_\psi(x,p) = \frac{1}{\pi\hbar} \int_{-\infty}^{\infty}ds\, e^{-2ips/\hbar} \psi(x+s)\psi^*(x-s).$$

Both formulas are correct. They imply different discrete frequency maps when $y$ and $s$ are sampled on different grids. Choose one form for the implementation and derive its momentum grid before writing code.

The notebook should use the $y$ form because it maps directly to an ordinary Fourier transform with conjugate variable $p/\hbar$.

Relative-Coordinate Correlation

For each phase-space position $x$, define

$$C_x(y) = \psi\left(x+\frac{y}{2}\right) \psi^*\left(x-\frac{y}{2}\right).$$

Then

$$W_\psi(x,p) = \frac{1}{2\pi\hbar} \mathcal F_y[C_x](p/\hbar).$$

The Wigner transform is therefore a family of one-dimensional Fourier transforms, one for each output position. Its cost is approximately

$$O(N_xN_y\log N_y)$$

when an FFT is used along the relative-coordinate axis.

The correlation probes off-diagonal coherence. For fixed $x$, large $\lvert y\rvert$ samples wavefunction values far apart. Truncating $y$ too early removes long-range coherence and produces ringing in momentum.

Discrete FFT Formula

Choose an evenly spaced relative grid

$$y_m = \left( m-\frac{N_y}{2} \right)\Delta y, \qquad m=0,\ldots,N_y-1,$$

with even $N_y$. The quadrature approximation is

$$\begin{aligned} W_{jk} &\approx \frac{\Delta y}{2\pi\hbar} \sum_{m=0}^{N_y-1} C_{x_j}(y_m) e^{-ip_ky_m/\hbar}. \end{aligned}$$

For the standard forward DFT, the physical momentum bins are

$$p_k = 2\pi\hbar\, \operatorname{fftfreq} \left( N_y,\Delta y \right),$$

followed by the same frequency shift used on the FFT output. Thus

$$\Delta p = \frac{2\pi\hbar}{N_y\Delta y},$$

and the momentum Nyquist scale is

$$p_{\rm Ny} = \frac{\pi\hbar}{\Delta y}.$$

The implementation sequence is:

1. place the centered $C_x(y_m)$ array into FFT order with an inverse shift;
2. apply the forward FFT;
3. shift the output and the frequency bins together;
4. multiply by $\Delta y/(2\pi\hbar)$.

A missing $\Delta y$, $2\pi$, or $\hbar$ changes both normalization and the momentum axis.

The Half-Grid Problem

The arguments

$$x+\frac{y}{2}, \qquad x-\frac{y}{2}$$

do not generally lie on the original wavefunction grid. There are three controlled options:

1. evaluate an analytic benchmark wavefunction directly at the shifted points;
2. evaluate the numerical wavefunction on a grid with half the output spacing;
3. interpolate the complex wavefunction and demonstrate interpolation convergence.

The second option avoids interpolation. If output positions are separated by $\Delta x$ and $\Delta y=\Delta x$, store $\psi$ on a fine grid of spacing $\Delta x/2$. Choose output $x_j$ at every second fine-grid point. Then

$$x_j\pm\frac{y_m}{2}$$

always lands on the fine grid.

Interpolating the density $\lvert\psi\rvert^2$ is not sufficient. The Wigner transform needs the complex amplitude and relative phase.

A Common Aliasing Trap

Another popular discretization samples the $s$ form at $s_m=m\Delta x$:

$$\begin{aligned} W(x_j,p) &\approx \frac{\Delta x}{\pi\hbar} \sum_m \psi_{j+m}\psi_{j-m}^* e^{-2ipm\Delta x/\hbar}. \end{aligned}$$

The factor of two in the exponential means the DFT momentum bins are

$$p_k = \frac{\pi\hbar k}{N_s\Delta x},$$

not the ordinary wavefunction-FFT bins

$$\frac{2\pi\hbar k}{N_s\Delta x}.$$

Assigning the ordinary bins doubles the physical momentum axis and aliases features. The $y$-grid implementation avoids this ambiguity by using the standard Fourier pair explicitly.

Gaussian Benchmark

Use the normalized packet

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

Its momentum width is

$$\sigma_p = \frac{\hbar}{2\sigma_x},$$

and its exact Wigner function is

$$\begin{aligned} W_G(x,p) &= \frac{1}{\pi\hbar} \exp\left[ - \frac{(x-x_0)^2}{2\sigma_x^2} \right. \\ &\qquad\left. - \frac{(p-p_0)^2}{2\sigma_p^2} \right]. \end{aligned}$$

Compare the complete complex numerical transform with this formula. The imaginary part should vanish to numerical tolerance, and the real part should converge pointwise and in an integrated norm.

Normalization and Marginals

On a rectangular phase-space grid, compute

$$\mathcal Z_W = \Delta x\,\Delta p \sum_{j,k}W_{jk}.$$

The target is

$$\mathcal Z_W=1.$$

The position marginal is

$$P_x(x_j) \approx \Delta p \sum_kW_{jk},$$

and should agree with

$$\lvert\psi(x_j)\rvert^2.$$

The momentum marginal is

$$P_p(p_k) \approx \Delta x \sum_jW_{jk}.$$

Compare it with an independently normalized Fourier transform of $\psi$, evaluated on the same physical momentum grid. Deriving both arrays from the same incorrectly scaled Wigner transform is not an independent check.

Reality, Purity, and Overlaps

For a Hermitian density operator, $W$ is real. Report

$$\epsilon_{\rm Im} = \frac{ \max_{j,k}\lvert\operatorname{Im}W_{jk}\rvert }{ \max_{j,k}\lvert\operatorname{Re}W_{jk}\rvert }.$$

For one degree of freedom,

$$2\pi\hbar \int dx\,dp\, W_\rho(x,p)^2 = \operatorname{Tr}\rho^2.$$

Thus a pure state satisfies

$$\mathcal P_W = 2\pi\hbar \Delta x\,\Delta p \sum_{j,k} W_{jk}^2 = 1.$$

For two pure states,

$$2\pi\hbar \int dx\,dp\, W_1W_2 = \lvert\langle\psi_1\vert\psi_2\rangle\rvert^2.$$

These identities are sensitive to normalization and phase-space cell area. They are valuable checks beyond the marginals.

Moments and Covariance

For Weyl-ordered polynomial observables, phase-space moments reproduce quantum expectations:

$$\langle x\rangle = \int dx\,dp\,xW,$$ $$\langle p\rangle = \int dx\,dp\,pW,$$

and

$$\frac12 \langle xp+px\rangle = \int dx\,dp\,xpW.$$

Construct the numerical covariance matrix

$$V = \begin{pmatrix} (\Delta x)^2 & C_{xp}\\ C_{xp} & (\Delta p)^2 \end{pmatrix}.$$

For the minimum-uncertainty Gaussian,

$$\det V = \frac{\hbar^2}{4}.$$

Compare these moments with direct wavefunction calculations. A Wigner array can pass normalization while using a shifted or reflected momentum convention; first moments catch that error.

Harmonic-Oscillator Evolution

Use a displaced Gaussian with

$$\sigma_x \ne \sqrt{\frac{\hbar}{2m\omega}}$$

so its Wigner ellipse is visibly squeezed relative to the oscillator ground state. Expand the wavefunction in the oscillator energy basis, evolve the coefficients by

$$c_n(t) = e^{-i\omega t(n+1/2)}c_n(0),$$

reconstruct $\psi(x,t)$, and compute $W_{\rm num}(x,p,t)$.

For the quadratic oscillator, the exact Wigner evolution is a classical phase-space rotation:

$$W(x,p,t) = W(x_{-t},p_{-t},0),$$

where

$$x_{-t} = x\cos\omega t - \frac{p}{m\omega}\sin\omega t,$$

and

$$p_{-t} = p\cos\omega t + m\omega x\sin\omega t.$$

Compare the numerically transformed evolved wavefunction with this rotated initial Wigner function. The test simultaneously checks basis evolution, complex wavefunction reconstruction, Wigner transformation, and phase-space orientation.

The center should follow

$$\begin{aligned} x_c(t) &= x_0\cos\omega t + \frac{p_0}{m\omega}\sin\omega t, \\ p_c(t) &= p_0\cos\omega t - m\omega x_0\sin\omega t. \end{aligned}$$

The covariance should rotate without changing its determinant.

Superposition-State Benchmark

Let $\psi_\pm$ be real minimum-uncertainty Gaussian packets of width $\sigma$ centered at

$$x=\pm\frac d2$$

with zero mean momentum. Their overlap is

$$S = \langle\psi_+\vert\psi_-\rangle = e^{-d^2/(8\sigma^2)}.$$

Define

$$\psi_{\rm cat} = \mathcal N \left( \psi_+ + e^{i\phi}\psi_- \right),$$

where

$$\mathcal N = \left[ 2(1+S\cos\phi) \right]^{-1/2}.$$

Its exact Wigner function is

$$\begin{aligned} W_{\rm cat}(x,p) &= \frac{\mathcal N^2}{\pi\hbar} e^{-2\sigma^2p^2/\hbar^2} \\ &\quad\times \left\{ e^{-(x-d/2)^2/(2\sigma^2)} + e^{-(x+d/2)^2/(2\sigma^2)} \right. \\ &\qquad\left. + 2e^{-x^2/(2\sigma^2)} \cos\left( \frac{dp}{\hbar}+\phi \right) \right\}. \end{aligned}$$

The first two terms are positive packet lobes. The last term is the interference contribution. It oscillates in momentum with fringe spacing

$$\Delta p_{\rm fringe} = \frac{2\pi\hbar}{d}.$$

The numerical momentum spacing must resolve this scale.

Odd Superposition and Exact Negativity

Choose

$$\phi=\pi.$$

The state is odd under parity. At the phase-space origin,

$$W_{\rm cat}(0,0) = - \frac{1}{\pi\hbar},$$

independent of $d$ and $\sigma$ when the state is normalized. This is a sharp pointwise benchmark for sign, normalization, phase convention, and interpolation.

Use

$$\sigma=0.55, \qquad d=4, \qquad \hbar=1$$

as the baseline. The negative interference bands should be clearly separated from the positive lobes.

Negative Volume

For a normalized continuous Wigner function, define

$$\mathcal N_W = \frac12 \left[ \int dx\,dp\,\lvert W\rvert - 1 \right].$$

On a numerical grid, use the equivalent signed form

$$\begin{aligned} \mathcal N_W^{\rm grid} &= \frac{\Delta x\,\Delta p}{2} \sum_{j,k} \left( \lvert W_{jk}\rvert-W_{jk} \right). \end{aligned}$$

Report $\mathcal Z_W$ separately rather than forcing it to one. For the Gaussian benchmark, $\mathcal N_W$ should converge to zero. Its residual value defines a numerical negativity floor. The superposition negativity is credible only when it remains well above that floor and converges under all relevant refinements.

Baseline Grids

Set

$$\hbar=m=\omega=1.$$

For the static Gaussian use

$$\sigma_x=0.7, \qquad x_0=1, \qquad p_0=0.8.$$

Choose

$$-8\le x\le8, \qquad \Delta x=\Delta y=\frac{1}{16}, \qquad N_y=1024.$$

Then

$$\Delta p = \frac{2\pi}{64} \approx 0.0982.$$

For harmonic evolution use a narrower initial width such as

$$\sigma_x=0.45$$

to make covariance rotation visible. For the odd superposition use the $\sigma$ and $d$ values above.

These values produce a useful first run for analytic wavefunction evaluation. Generic numerical wavefunctions still require independent interpolation, basis-cutoff, and boundary studies.

Notebook Workflow

Organize the notebook into these stages:

1. define the Fourier and Wigner conventions;
2. construct the $x$, $y$, and derived $p$ grids;
3. assert the FFT frequency spacing and ordering;
4. evaluate the Gaussian correlation array;
5. apply the shifted FFT with the physical prefactor;
6. compare with the analytic Gaussian Wigner function;
7. validate reality, normalization, both marginals, moments, covariance, purity, and zero negativity;
8. evolve a squeezed Gaussian in the oscillator basis;
9. compare its numerical Wigner function with exact phase-space rotation;
10. evaluate the odd superposition and compare with its analytic Wigner function;
11. measure $W(0,0)$, fringe spacing, and negative volume;
12. repeat the grid, domain, relative-range, and interpolation refinements;
13. run assertions before producing final plots.

Record array shapes, endpoint conventions, shift order, FFT normalization, physical spacings, interpolation method, and all tolerances.

Validation Checks

The minimum validation suite is:

Check	Target
Grid map	$\Delta p=2\pi\hbar/(N_y\Delta y)$ and shifted ordering agree
Reality	imaginary residual approaches zero
Normalization	$\mathcal Z_W\to1$
Position marginal	agrees with $\lvert\psi(x)\rvert^2$
Momentum marginal	agrees with an independent wavefunction FFT
Gaussian formula	full Wigner array converges to $W_G$
First moments	reproduce $x_0$ and $p_0$
Covariance	agrees with direct wavefunction moments
Purity	$\mathcal P_W\to1$ for every pure benchmark
Gaussian negativity	converges to zero and defines the error floor
Harmonic transport	agrees with exact phase-space rotation
Phase-space determinant	remains $\hbar^2/4$ for the pure Gaussian
Odd superposition	$W(0,0)\to-1/(\pi\hbar)$
Fringe spacing	agrees with $2\pi\hbar/d$
Superposition negativity	stabilizes above the numerical floor

For harmonic evolution, also check that a full oscillator period returns the Wigner function to itself. This test is insensitive to the wavefunction’s global phase, as a density-operator representation should be.

Convergence Study

Vary the numerical scales independently:

1. Position range. Increase the wavefunction domain and monitor boundary amplitude.
2. Output spacing. Refine $\Delta x$ while keeping physical ranges fixed.
3. Relative range. Increase $N_y\Delta y$ at fixed $\Delta y$ to refine momentum spacing.
4. Relative spacing. Decrease $\Delta y$ at fixed relative range to increase momentum bandwidth.
5. Interpolation. Compare analytic evaluation, half-grid sampling, and at least one converged complex interpolation method.
6. Oscillator basis. Increase the number-state cutoff used for time evolution.
7. Cat separation. Increase $d$ and confirm that the required momentum resolution tracks the shrinking fringe spacing.

Zero padding increases the density of sampled momentum bins but does not restore correlation data that were truncated or wavefunction structure that was under-resolved. Distinguish interpolation of a transform from increased physical resolution.

Visualization

Use a signed diverging color map centered exactly at zero. Set symmetric color limits when comparing positive and negative amplitudes, and include a color bar with physical units. Plot $x$ and $p$ on their actual derived grids.

For the Gaussian, show contours and marginal panels. For oscillator evolution, use equal-aspect scaled coordinates

$$Q=\sqrt{\frac{m\omega}{\hbar}}x, \qquad P=\frac{p}{\sqrt{m\hbar\omega}},$$

so a phase-space rotation is geometrically visible. For the superposition, include a one-dimensional cut through $x=0$ that resolves the predicted cosine fringes.

Do not clip negative values, smooth the Wigner array, or use independent color scales across convergence frames without saying so. Each can hide the numerical signature being tested.

Expected Results

For the proposed analytic grids:

• the Gaussian transform agrees with the exact array to near floating-point precision;
• normalization, purity, and both marginals are correspondingly accurate;
• Gaussian negative volume is at the numerical floor;
• the oscillator Wigner ellipse rotates without changing area;
• the transformed evolved wavefunction agrees with the backward-rotated initial Wigner function;
• the odd superposition reaches $W(0,0)=-1/\pi$ for $\hbar=1$;
• its negative volume stabilizes near $0.28$ for the baseline $d$ and $\sigma$;
• coarsening the momentum grid first corrupts the interference fringes;
• truncating the relative-coordinate range produces momentum ringing;
• using the wrong FFT momentum map shifts or doubles all momentum-space features.

The near-machine-precision static benchmarks rely on analytic evaluation at shifted points. A generic interpolated wavefunction should be held to a convergence-based tolerance instead.

Common Numerical Failures

• Using the $s$-grid formula with ordinary FFT momentum bins. The factor of two changes the physical frequency map.
• Interpolating probability density instead of the complex wavefunction.
• Shifting the FFT output without shifting the frequency bins, or vice versa.
• Dropping $\Delta y/(2\pi\hbar)$.
• Including duplicate periodic endpoints in an FFT grid.
• Truncating $C_x(y)$ while it is still appreciable. This causes momentum ringing.
• Calling tiny negative Gaussian pixels physical negativity. Establish the refinement floor first.
• Renormalizing the Wigner array to hide a prefactor or domain error.
• Checking only the position marginal. A wrong momentum axis can still leave it plausible.
• Computing purity without the phase-space cell area and $2\pi\hbar$ factor.
• Using zero padding as a substitute for physical resolution.
• Smoothing interference fringes before measuring negative volume.
• Plotting unequal scaled axes and misreading rotation as squeezing.
• Treating a positive Gaussian Wigner function as proof that the state is classical.

Extensions

• Compute Wigner functions for oscillator number states and verify alternating parity at the origin.
• Compare coherent superpositions with incoherent mixtures of the same two packets.
• Apply Gaussian coarse graining and observe how interference negativity is suppressed.
• Reconstruct Wigner functions from simulated quadrature marginals by an inverse Radon transform.
• Evolve the superposition under the harmonic oscillator and verify rigid fringe rotation.
• Introduce a quartic potential and compare classical transport with Moyal corrections.

Cross-Links

• Wigner Function
• Marginals and Quasi-Probabilities
• Phase-Space Dynamics
• Gaussian States and Wigner Functions
• Coherent States in Phase Space
• Phase-Space Conventions
• Gaussian Wave Packets
• Harmonic-Oscillator Propagator Notebook
• Fast Fourier Transform
• Convergence Tests
• Validation Tests

References

• E. Wigner, “On the Quantum Correction For Thermodynamic Equilibrium,” Physical Review 40, 749–759 (1932).
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution Functions in Physics: Fundamentals,” Physics Reports 106, 121–167 (1984), doi:10.1016/0370-1573(84)90160-1.
• W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.
• U. Leonhardt, Measuring the Quantum State of Light, Cambridge University Press, 1997.
• W. B. Case, “Wigner Functions and Weyl Transforms for Pedestrians,” American Journal of Physics 76, 937–946 (2008).
• NumPy Developers, Discrete Fourier Transform documentation, consulted for FFT normalization, frequency ordering, and shift conventions.

Exercises

1. Derive the FFT momentum grid

Starting from the discrete $y$ transform, derive $\Delta p$ and $p_{\rm Ny}$.

Solution

The DFT phase is

$$e^{-2\pi ikm/N_y}.$$

The Wigner phase for grid points $y_m=m\Delta y$ is

$$e^{-ip_km\Delta y/\hbar}.$$

Equating exponents gives

$$\frac{p_k\Delta y}{\hbar} = \frac{2\pi k}{N_y},$$

so

$$p_k = \frac{2\pi\hbar k}{N_y\Delta y}.$$

Adjacent bins differ by

$$\Delta p = \frac{2\pi\hbar}{N_y\Delta y}.$$

The largest represented angular frequency has magnitude approximately $\pi/\Delta y$, giving

$$p_{\rm Ny} = \frac{\pi\hbar}{\Delta y}.$$

2. Prove the position marginal

Integrate the defining Wigner transform over momentum.

Solution

Start from

$$\begin{aligned} \int dp\,W(x,p) &= \frac{1}{2\pi\hbar} \int dy\, \psi\left(x+\frac y2\right) \psi^*\left(x-\frac y2\right) \\ &\quad\times \int dp\,e^{-ipy/\hbar}. \end{aligned}$$

The Fourier identity is

$$\frac{1}{2\pi\hbar} \int dp\,e^{-ipy/\hbar} = \delta(y).$$

The $y$ integral therefore evaluates the product at $y=0$:

$$\int dp\,W(x,p) = \lvert\psi(x)\rvert^2.$$

3. Derive the odd-cat value at the origin

Use the analytic superposition Wigner function with $\phi=\pi$ to show that $W(0,0)=-1/(\pi\hbar)$.

Solution

At $x=p=0$, each positive lobe contributes

$$e^{-d^2/(8\sigma^2)} = S.$$

The interference term contributes

$$2\cos\pi=-2.$$

Thus

$$W(0,0) = \frac{\mathcal N^2}{\pi\hbar} (2S-2).$$

For $\phi=\pi$,

$$\mathcal N^2 = \frac{1}{2(1-S)}.$$

Therefore

$$W(0,0) = \frac{2(S-1)} {2(1-S)\pi\hbar} = - \frac{1}{\pi\hbar}.$$

4. Explain harmonic backward rotation

Why is the evolved Wigner function evaluated at the inverse classical flow rather than the forward coordinates?

Solution

The Wigner value is transported along a trajectory. If a phase-space point $\mathbf z_0$ moves to

$$\mathbf z(t)=S(t)\mathbf z_0,$$

then the density at the final point equals the initial density at its preimage:

$$W(\mathbf z,t) = W(S(-t)\mathbf z,0).$$

The arguments $(x_{-t},p_{-t})$ are precisely $S(-t)(x,p)$. Using the forward map inside $W_0$ rotates the pattern in the wrong direction.

5. Show that purity equals one for the Gaussian

Evaluate $2\pi\hbar\int W_G^2\,dx\,dp$.

Solution

Using

$$W_G = \frac{1}{\pi\hbar} \exp\left[ - \frac{(x-x_0)^2}{2\sigma_x^2} - \frac{(p-p_0)^2}{2\sigma_p^2} \right],$$

one obtains

$$\begin{aligned} \int dx\,dp\,W_G^2 &= \frac{1}{\pi^2\hbar^2} \left( \sqrt\pi\,\sigma_x \right) \left( \sqrt\pi\,\sigma_p \right) \\ &= \frac{\sigma_x\sigma_p}{\pi\hbar^2}. \end{aligned}$$

For a minimum-uncertainty Gaussian,

$$\sigma_x\sigma_p = \frac{\hbar}{2},$$

so

$$2\pi\hbar \int dx\,dp\,W_G^2 = 1.$$

6. Resolve the interference fringes

Suppose a cat state has packet separation $d$. What relative-coordinate range is needed to sample at least $q$ momentum points per fringe?

Solution

The fringe spacing is

$$\Delta p_{\rm fringe} = \frac{2\pi\hbar}{d}.$$

The FFT momentum spacing is

$$\Delta p = \frac{2\pi\hbar}{Y}, \qquad Y=N_y\Delta y.$$

At least $q$ points per fringe requires

$$\Delta p \le \frac{\Delta p_{\rm fringe}}{q}.$$

Substitution gives

$$\frac{2\pi\hbar}{Y} \le \frac{2\pi\hbar}{qd},$$

or

$$Y\ge qd.$$

This controls momentum spacing. The separate condition $\Delta y$ small enough to cover the required momentum bandwidth must also be checked.