Polar harmonic expansion

Function space and convention

Let \(f(r,\theta)\) be a complex scalar field on a disk \(0\le r\le R\), \(0\le\theta<2\pi\), with finite polar \(L^2\) norm

\[ \|f\|_{L^2(D_R)}^2= \int_0^R\int_0^{2\pi}|f(r,\theta)|^2\,r\,d\theta\,dr<\infty. \]

For almost every fixed \(r\), the angular function belongs to \(L^2(S^1)\) and therefore has a Fourier expansion

\[ f(r,\theta)=\sum_{m\in\mathbb Z}\rho_m(r)e^{im\theta}, \]

with PETAL2D’s continuum normalization

\[ \rho_m(r)=\frac{1}{2\pi}\int_0^{2\pi} f(r,\theta)e^{-im\theta}\,d\theta. \]

This is the continuum convention approximated by fft(f_polar)/Ntheta.

Physical meaning of \(m\)

The angular momentum operator in two-dimensional polar coordinates is

\[ L_z=-i\hbar\frac{\partial}{\partial\theta}. \]

Because

\[ L_z e^{im\theta}=m\hbar e^{im\theta}, \]

the \(e^{im\theta}\) channels are angular-momentum eigenfunctions. If \(f\) is a complex wavefunction, \(\rho_m(r)\) resolves its \(L_z\) content while retaining radial information.

For a density or other real scalar observable, the same basis is still mathematically natural, but \(m\) labels angular shape harmonics, not necessarily a quantum number of an underlying state.

For a normalized complex wavefunction, the fraction \(P_m/\sum_nP_n\) is the probability weight in the \(L_z=m\hbar\) angular sector after the radial degree of freedom is summed over. This interpretation is proved explicitly in Proofs.

Mode power

Define the radial power of mode \(m\) as

\[ P_m=\int_0^R|\rho_m(r)|^2r\,dr. \]

Angular Parseval gives

\[ \frac{1}{2\pi}\int_0^{2\pi}|f(r,\theta)|^2d\theta =\sum_m|\rho_m(r)|^2. \]

Integrating over \(r\) yields

\[ \|f\|_{L^2(D_R)}^2=2\pi\sum_mP_m. \]

Thus the fractional power

\[ w_m=\frac{P_m}{\sum_n P_n} \]

is a natural dimensionless measure of angular content.

Why retain radial profiles?

A single number \(P_m\) says how important a harmonic is globally. The function \(\rho_m(r)\) says where it lives. Two fields can have identical angular power fractions but very different radial structures, nodes, shell locations, or localization lengths. PETAL2D keeps both levels of information.

Point-group fingerprints

If a scalar field is exactly invariant under a \(2\pi/n\) rotation,

\[ f(r,\theta+2\pi/n)=f(r,\theta), \]

then only harmonics with \(m\) divisible by \(n\) may be nonzero. More generally, if a complex field transforms with a one-dimensional rotational character

\[ f(r,\theta+2\pi/n)=e^{i\ell 2\pi/n}f(r,\theta), \]

then nonzero harmonics satisfy

\[ m\equiv \ell \pmod n. \]

This makes PETAL2D useful for diagnosing local crystal-field mixing and rotational symmetry around a site. The formal proof is in Proofs.

Warning

A spectrum compatible with a point group is evidence of angular structure, not by itself a proof of full spatial symmetry. Radial dependence, the chosen origin, additional reflections, layer/orbital labels, and other degrees of freedom may matter.