Example 1: isotropic QHO ground-state density

The two-dimensional isotropic harmonic-oscillator ground-state wavefunction is rotationally symmetric. Its probability density may be written, up to units and normalization, as

\[ n(r)=\frac{1}{\pi}e^{-r^2}. \]

There is no angular dependence, so theory predicts

\[ \rho_0(r)=n(r),\qquad \rho_{m\ne0}(r)=0. \]

Run:

python examples/qho_ground_state.py

=== 2D QHO ground-state probability density ===
domain_consistency: 0.99941502
selected_pairs: [(0,)]
retained_modes: [0]
reconstruction_error_percent: 1.40994e-14
target_reached: True
     m           power      fraction   r_power_support   r_amplitude_support      r_cutoff  retained
----------------------------------------------------------------------------------------------------
     0    2.531548e-02  1.000000e+00           2.63121               2.62905       2.63121      True

The \(m=0\) result also checks radial integration, reconstruction, and the cutoff-radius diagnostic for a smooth monotone tail.

Note

Because the input here is a density, the spectrum describes density anisotropy. An excited wavefunction \(\psi\propto e^{im\theta}\) can carry nonzero angular momentum while \(|\psi|^2\) remains isotropic.

The isotropic density is entirely \(m=0\) within numerical precision.

Original field and retained-mode reconstruction on a common color scale.