Visualization Gallery

This gallery records visual assets for symmetry, spin, angular momentum, and geometric phase. Its purpose is not decoration. A useful visualization states what mathematical object is being shown, which convention is being used, what is schematic versus computed, and where the source or validation lives.

The companion page Computational Notebooks records planned notebook families. Notebook-generated figures should not enter this gallery until their source notebook includes validation cells and reproducibility metadata.

Current Static Gallery

The current entries are source-tracked SVG assets that already support nearby canonical pages. They live under public/figures/wave-mechanics/ because they were created for canonical-system pages, but they are also useful here as cross-volume visual references. Future native assets for this volume should use public/figures/symmetry/ and public/figures-src/symmetry/.

Bloch-sphere representation of a two-level state. Use this visual for state rays and spin-$1/2$ expectation directions, not for literal particle motion. The canonical discussion is Bloch Sphere.

Low-order spherical-harmonic angular shapes. The figure is a recognition aid; calculations should use the complex normalized $Y_\ell^m$ convention stated in Spherical Harmonics Quick Reference.

Two paths enclosing excluded magnetic flux acquire a relative Aharonov–Bohm phase. The visual emphasizes holonomy in the accessible region, not a local force along the paths. See Aharonov–Bohm Effect.

Asset	Source	Status	Canonical page
/figures/wave-mechanics/bloch-sphere-two-level.svg	/figures-src/wave-mechanics/bloch-sphere-two-level.tex	committed static SVG; schematic convention figure	Bloch Sphere
/figures/wave-mechanics/spherical-harmonics-angular-shapes.svg	/figures-src/wave-mechanics/spherical-harmonics-angular-shapes.tex	committed static SVG; shape-recognition figure	Spherical Harmonics
/figures/wave-mechanics/aharonov-bohm-two-path-phase.svg	/figures-src/wave-mechanics/aharonov-bohm-two-path-phase.tex	committed static SVG; conceptual phase-holonomy figure	Aharonov–Bohm Effect

Candidate Visualizations

The following candidates should be added only after the source figure or notebook exists and passes the admission checklist.

Candidate	What it should show	Source expectation	Required warning or validation
Spin rotation trajectories	Bloch vector moving under a stated Hamiltonian	notebook or animation source	norm and Bloch-vector length conservation
Angular momentum cones	fixed $j,m$ precession-style schematic	TikZ source	semiclassical picture; not a literal orbit
Coupled angular momentum vector model	allowed $J$ values from $j_1$ and $j_2$	TikZ or notebook	vector model is mnemonic; exact content is representation theory
Berry phase as solid angle	loop on the parameter sphere and enclosed $\Omega$	TikZ or validated notebook	sign tied to Hamiltonian and eigenstate convention
Aharonov–Bohm ring interference	flux-dependent fringe or ring-spectrum shift	notebook source	periodicity under $\Phi_B\mapsto\Phi_B+\Phi_0$
Level splitting under symmetry breaking	degeneracy lifting as a parameter is turned on	notebook source	conserved labels and broken symmetries stated
Zeeman multiplet diagram	$m$-dependent shifts in a magnetic field	TikZ or notebook	sign of coupling and $g$ factor convention stated
Time-reversal action on spinors	spin reversal plus complex conjugation	schematic or exact notebook	antiunitary nature and $T^2$ sign stated

Admission Checklist

A visualization belongs here only when it has:

• a stable asset path under public/figures/;
• a source path under public/figures-src/ or a source notebook under notebooks/;
• alt text that states the physical content;
• a caption that names the plotted object and convention;
• a validation statement or explicit schematic label;
• a link to the canonical page where the physics is explained;
• no hidden dependence on a gauge, phase, basis ordering, or sign convention.

For notebook-generated figures, use the status labels from Reproducibility Status. A plot with no validation cell should remain a candidate.

Caption Rules

Captions for this volume should answer:

1. Is the figure a state, operator, expectation value, probability density, spectrum, phase, or schematic?
2. Which convention is used for basis order, phase, gauge, or orientation?
3. Which parameters are fixed?
4. What mathematical fact or physical lesson should the reader see?
5. What check supports the figure if it is generated from a computation?

For example, a Berry-phase solid-angle plot should state the loop orientation and the convention for the eigenstate. A spherical-harmonic plot should state whether it shows complex $Y_\ell^m$, a real linear combination, the sign, or a probability density.

Accessibility and Design

Use labels and line styles in addition to color. Do not rely on red/blue signs alone for positive and negative lobes. For dense diagrams, keep mathematical labels short and place detailed interpretation in the caption.

Avoid visual metaphors that overpower the exact statement. Angular momentum cones, coupled-vector cartoons, and spin arrows are useful mnemonics, but the caption must say when the exact object is an operator algebra or Hilbert-space ray.

Common Mistakes

• Treating the Bloch sphere as physical space.
• Showing real spherical harmonics while citing complex $Y_\ell^m$ formulas without warning.
• Drawing spin as a tiny rotating ball.
• Plotting a gauge-dependent quantity without naming the gauge.
• Using a vector-model angular momentum diagram as if it were an exact derivation.
• Exporting a notebook-generated figure before convergence or normalization checks.
• Omitting parameter values from a level-splitting or Berry-phase plot.

Cross-Links

• Computational Notebooks
• Bloch Sphere
• Spherical Harmonics Quick Reference
• Angular Momentum Problems
• Berry Phase Problems
• Aharonov–Bohm Effect
• Degeneracy Lifting
• Reproducibility Status

References

• E. Tufte, The Visual Display of Quantitative Information, 2nd ed., Graphics Press, 2001.
• R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, 1988.
• A. Shapere and F. Wilczek, eds., Geometric Phases in Physics, World Scientific, 1989.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. A plot shows a spin arrow moving on a sphere but does not say whether it is a Bloch vector or a physical-space trajectory. Why should it remain out of the gallery?

Solution

The interpretation is ambiguous. A Bloch vector represents expectation values or a ray in a two-level Hilbert space; it is not a literal particle trajectory. The caption must state the object being plotted and the Hamiltonian or transformation that generates the motion.

2. A spherical-harmonic figure uses real lobes but the nearby text quotes complex $Y_\ell^m$ formulas. What must the caption explain?

Solution

It must state that the plotted lobes are real linear combinations of complex spherical harmonics, not the complex $L_z$ eigenfunctions themselves. It should also identify the sign, normalization, and whether the surface radius represents amplitude, signed amplitude, or probability density.