Example 2: a pure angular-momentum state

Consider a separable localized wavefunction

\[ \psi_m(r,\theta)=r^{|m|}e^{-\alpha r^2}e^{im\theta}. \]

The factor \(r^{|m|}\) makes the field regular at the origin, while the angular factor is an exact eigenfunction of the planar angular-momentum operator:

\[ L_z\psi_m=m\hbar\psi_m. \]

The executable example uses \(m=4\):

python examples/angular_momentum_state.py

=== Complex localized angular-momentum state (m=4) ===
domain_consistency: 1.00000000
selected_pairs: [(4,)]
retained_modes: [4]
reconstruction_error_percent: 8.55582e-14
target_reached: True
     m           power      fraction   r_power_support   r_amplitude_support      r_cutoff  retained
----------------------------------------------------------------------------------------------------
     4    2.231213e+00  1.000000e+00           4.09379               4.24121       4.24121      True

PETAL2D therefore returns a single complex channel (4,). The power-ranked harmonic panel contains one categorical bar labelled 4. It does not create an empty linear axis from \(-4\) to \(4\). The same presentation remains compact even for a pure high-order harmonic such as \(m=100\).

Rotating the state by \(\phi\) multiplies \(\rho_4\) by \(e^{-4i\phi}\) but leaves \(P_4\) unchanged. Coefficient phase therefore carries orientation information, whereas harmonic power does not.

The \(m=4\) channel carries all angular power.

Magnitude of the original complex field and its reconstruction.