Applications and interpretation patterns

PETAL2D is a general angular-analysis tool, but its interpretation depends on what the input field represents. The same mathematical spectrum can carry different physical meaning for a wavefunction, a density, an order parameter, or an image.

Complex wavefunctions and angular momentum

For a normalized complex wavefunction on the analyzed disk, with angular momentum defined about the chosen PETAL2D origin, the power fraction

\[ w_m=\frac{P_m}{\sum_nP_n} \]

is exactly the probability weight of the \(L_z=m\hbar\) angular sector within that disk, with radial degrees of freedom left unresolved. This follows because \(e^{im\theta}\) diagonalizes \(L_z\) and the angular sectors are orthogonal.

This is useful for planar model wavefunctions, defect states, excitonic relative-coordinate states, and localized continuum eigenmodes.

Atomic-orbital and Wannier character

For a localized orbital centered on an atomic, molecular, or Wannier site, the dominant \(m\) channels provide an angular fingerprint. Familiar real orbitals appear as conjugate combinations: \(p_x\) contains \(m=\pm1\), while \(d_{x^2-y^2}\) contains \(m=\pm2\).

In a crystal, however, \(m\) is usually a basis label, not an exact isolated-atom quantum number. Crystal fields can mix angular channels allowed by the local point-group symmetry. PETAL2D is especially useful for quantifying that mixing rather than forcing one orbital label.

Local crystal symmetry

A converged spectrum can expose rotational selection rules. An exactly \(C_3\)-invariant scalar field about the correct center contains only \(m=0,\pm3,\pm6,\ldots\). A field transforming under a nontrivial one-dimensional \(C_n\) character occupies one congruence class \(m\equiv\ell\pmod n\).

This makes the method useful for localized crystal states, high-symmetry regions, and local orbital textures. Weak symmetry-incompatible modes should be convergence-tested before being interpreted as symmetry breaking because Cartesian-to-polar interpolation can generate small leakage.

Densities and scalar observables

For charge density, probability density, a scalar spin-density component, or another real observable, PETAL2D describes shape anisotropy. It does not recover phase information that is absent from the observable. For example, \(|e^{im\theta}|^2=1\), so an angular-momentum eigenstate can have an isotropic probability density.

Complex order parameters and vortices

A complex scalar order parameter with phase winding \(\Delta(r,\theta)\sim g(r)e^{im\theta}\) appears naturally as a dominant complex \(m\) channel. PETAL2D can separate mixed windings while retaining each radial envelope. This can be useful for superfluid/superconducting order parameters, scalar wave vortices, and related 2D fields.

Photonic, phononic, acoustic, and PDE modes

Any localized scalar eigenmode of a two-dimensional wave or differential equation can be analyzed in the same way. The angular spectrum can compare mode families, symmetry breaking, defect-induced mixing, or changes in radial localization across parameters.

Scientific imaging and morphology

For a localized scalar intensity map, PETAL2D can serve as an interpretable angular descriptor. Unlike a generic image feature vector, the output retains a direct meaning in terms of angular harmonics and radial profiles. This is most appropriate when a physically meaningful center exists.

When PETAL2D is the wrong tool

Use a different representation when the problem is intrinsically three-dimensional, lacks a meaningful center, is dominated by translation/plane-wave structure, or requires coupled vector/tensor symmetry analysis. A Cartesian FFT, spherical harmonics, or problem-specific symmetry decomposition may then be more natural.