Computational Notebooks

This page is the volume-specific roadmap for reproducible computational notebooks in symmetry, spin, angular momentum, and geometric phase. It is an index and admission contract, not a substitute for the general Notebook Index or How to Use Computational Notebooks.

As of this review, no notebook family under notebooks/symmetry/ is promoted as a reproduced artifact. The entries below are planned notebook families with validation requirements. A notebook should not be cited as evidence, used to generate a figure, or treated as a reference artifact until it has a recorded reproducibility status.

Scope

Computational notebooks in this volume should make transformation laws visible:

spin rotations as $SU(2)$ matrices and Bloch-sphere motion;
angular-momentum addition as a unitary basis change;
spherical harmonics as orthonormal angular functions with phase conventions;
Zeeman and symmetry-breaking terms as controlled degeneracy splitting;
Berry phase as holonomy after dynamical phase removal;
Aharonov–Bohm phase as flux-dependent loop holonomy;
magnetic translations and Landau-level degeneracy as flux-counting structure;
time reversal as antiunitary action, not an ordinary matrix multiplication.

The notebook should answer one question at a time. A plot or animation is acceptable only when the calculation behind it is reproducible and validated.

Directory Plan

Use a dedicated volume directory:

notebooks/symmetry/
  spin-rotations/
  angular-momentum-addition/
  spherical-harmonics/
  zeeman-splitting/
  berry-phase/
  aharonov-bohm/
  magnetic-geometry/
  discrete-symmetries/

Each subdirectory should include an opening README or notebook cell stating shared units, package expectations, runtime, and validation philosophy. Notebook paths should be relative to the repository root.

Notebook Contract

Every notebook promoted from planned to reproduced should state:

purpose and canonical pages used;
Hamiltonian, operator, or transformation being computed;
units and dimensionless conventions;
basis ordering and tensor-product ordering where relevant;
phase, gauge, and Condon–Shortley conventions where relevant;
numerical method or exact finite-dimensional construction;
parameters and tolerances;
validation checks with pass/fail criteria;
environment and package versions;
reproducibility status and last run date;
known limitations and failure modes.

If a notebook generates a figure, the notebook must also state the plotted quantity, axes, units or dimensionless variables, parameter values, and export path.

Planned Notebook Families

Notebook family	Canonical pages	Method	Required validation	Status
Spin rotations and Bloch sphere	Spin Rotations, Bloch Sphere	exact $2\times2$ matrices and Bloch-vector maps	norm conservation, $\lvert\mathbf r\rvert$ preservation, $2\pi$ spinor sign, composition checks	`planned`
Addition of angular momentum	Clebsch–Gordan Coefficients, Angular Momentum Problems	finite matrices, ladder construction, optional exact symbolic checks	unitary coupled-basis matrix, dimension count, $M=m_1+m_2$ , known spin- $1/2$ singlet-triplet benchmark	`planned`
Spherical harmonics visualization	Spherical Harmonics, Spherical Harmonics Quick Reference	analytic functions on angular quadrature grids	normalization with $d\Omega$ , orthogonality, conjugation rule, parity $(-1)^\ell$	`planned`
Zeeman splitting simulation	Spin in Magnetic Fields, Degeneracy Lifting	finite spin Hamiltonians and parameter sweeps	linear weak-field shifts, preserved good quantum numbers, agreement with analytic spin- $1/2$ result	`planned`
Berry phase for rotating spin	Berry Phase for Spin-1/2, Berry Phase Problems	adiabatic time evolution and phase extraction	norm conservation, dynamical phase subtraction, convergence to $-\Omega/2$ as traversal slows	`planned`
Aharonov–Bohm interference phase	Aharonov–Bohm Effect, Particle on a Ring	flux-threaded ring or two-path phase model	$\Phi_0=h/\lvert q\rvert$ periodicity, winding-number phase, gauge-equivalent spectra	`planned`
Landau-level degeneracy visualization	Magnetic Translations, Landau Levels	finite geometry or Landau-gauge oscillator checks	level spacing, flux degeneracy scaling, boundary-condition caveats	`planned`
Time-reversal operator for spin systems	Time Reversal for Spin-1/2 Particles, Kramers Degeneracy	antiunitary action implemented as matrix plus complex conjugation	$T^2=-I$ for spin- $1/2$ , $T^2=+I$ for integer spin examples, Kramers-pair check	`planned`

The spherical-harmonics and Landau-level rows overlap with canonical-system notebooks. When a canonical-system notebook already owns the numerical artifact, this page should link to it instead of duplicating it.

Validation Patterns

Use exact finite-dimensional checks whenever possible. For spin rotations, verify

U^\dagger U=I

and compare the transformed Bloch vector with the corresponding $SO(3)$ rotation. For time evolution, verify norm conservation:

\left| 1-\langle\psi(t)|\psi(t)\rangle \right| < \epsilon_{\mathrm{norm}}.

For angular-momentum addition, the coupled-basis matrix $C$ must satisfy

C^\dagger C=I,

and the dimensions must match:

(2j_1+1)(2j_2+1) = \sum_{J=\lvert j_1-j_2\rvert}^{j_1+j_2} (2J+1).

For spherical harmonics, numerical quadrature should check

\int Y_\ell^m(\Omega)^* Y_{\ell'}^{m'}(\Omega) d\Omega = \delta_{\ell\ell'}\delta_{mm'}.

For Berry phase, run at multiple traversal times and verify convergence of the extracted geometric phase while the dynamical phase changes. For Aharonov–Bohm calculations, verify periodicity under

\Phi_B\mapsto\Phi_B+\Phi_0.

Status Labels

Use the repository-wide labels from Reproducibility Status. For this page:

Local use	Meaning
`planned`	notebook family is specified but no promoted artifact exists
`draft`	notebook exists but lacks complete validation or environment record
`reproduced`	notebook reran successfully with declared checks passing
`reproduced_with_warnings`	checks pass but warnings or limitations must be visible
`broken`	expected run or validation fails
`conceptual_only`	explanatory notebook without numerical-evidence status

Do not use an exploratory notebook as support for a page claim unless the page explicitly labels it exploratory.

Figure Export Policy

A notebook-generated figure can enter Visualization Gallery only after:

the source notebook has a validation cell;
the exported asset path is recorded;
the caption states the plotted quantity and parameters;
gauge-dependent quantities are labeled as such;
the figure is regenerated after substantial notebook changes.

For example, a Bloch-sphere trajectory should state the Hamiltonian, initial state, time interval, and whether the plotted curve is a state ray, a Bloch vector, or a spin expectation value.

Common Mistakes

Publishing visual output before validation cells are present.
Hiding a gauge or phase convention inside code.
Comparing Clebsch–Gordan coefficients from different conventions without a sign check.
Treating an antiunitary time-reversal operator as a plain unitary matrix.
Forgetting that a Berry phase notebook must remove the dynamical phase.
Using a finite-size magnetic-geometry model without stating boundary conditions.
Marking a notebook reproduced without recording the environment and run date.

Cross-Links

References

Project Jupyter, Jupyter Documentation.
C. R. Harris et al., “Array programming with NumPy,” Nature 585, 357–362, 2020.
L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.
J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.
M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45–57, 1984.
J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.