{
  "release": "0.6.0",
  "bounded_question": "What conserved flow and observable harmonics are implied by the existing hydrogen wave packet, and which density beats are removed by electric-dipole selection rules?",
  "model": "ideal fixed-proton non-relativistic Coulomb hydrogen with A=0",
  "written_mathematics": {
    "continuity_equation": "partial_t rho + div(j) = 0",
    "probability_current_A_zero": "j = (hbar/(2 m i)) (psi* grad psi - psi grad psi*)",
    "minimal_coupling_not_yet_simulated": "H = (-i hbar grad - q A)^2/(2 m) + q phi",
    "electric_dipole_observable": "d(t) = -e <psi(t)|r|psi(t)>"
  },
  "one_s_plus_two_pz": {
    "energy_gap_hartree": 0.375,
    "beat_frequency_hz": 2467381470187498.5,
    "beat_period_attoseconds": 405.28795894864567,
    "z_matrix_element_exact_a0": 0.7449355390278031,
    "z_matrix_element_grid_a0": 0.744932784051341,
    "equal_superposition_dipole_amplitude_C_m": 6.315826923825808e-30,
    "equal_superposition_dipole_amplitude_debye": 1.8934378181775633,
    "ideal_spontaneous_e1_rate_s_minus_1": 626831502.9564015,
    "ideal_spontaneous_lifetime_ns": 1.595325051921575,
    "grid_z_expectation_at_phase_zero_a0": 0.7449327840513413,
    "grid_integrated_jz_at_phase_pi_over_2_a0_per_atomic_time": -0.27950009342972815,
    "expected_integrated_jz_a0_per_atomic_time": -0.27935082713542614
  },
  "selection_rule_counterexamples": {
    "one_s_plus_two_s": "The local density has a beat, but <1s|z|2s>=0 by parity, so no one-photon E1 dipole harmonic appears.",
    "one_s_two_s_grid_z_matrix_element_a0": 2.4395416242661765e-17,
    "one_s_plus_three_s": "The three-state density contains the 1s<->3s gap, but the z-polarized electric-dipole signal does not."
  },
  "three_state_harmonic_catalog": [
    {
      "pair": "1s<->2p_z",
      "energy_gap_hartree": 0.375,
      "frequency_hz": 2467381470187498.5,
      "angular_frequency_rad_s": 1.5503015000689256e+16,
      "vacuum_wavelength_nm": 121.50227341101758,
      "density_harmonic_present": true,
      "z_matrix_element_a0": 0.7449355390278031,
      "equal_three_state_z_amplitude_a0": 0.49662369268520207,
      "electric_dipole_harmonic_present": true,
      "e1_angular_rule": "Delta l = +/-1 and Delta m = 0 for z polarization"
    },
    {
      "pair": "1s<->3s",
      "energy_gap_hartree": 0.4444444444444444,
      "frequency_hz": 2924303964666664.5,
      "angular_frequency_rad_s": 1.83739437045206e+16,
      "vacuum_wavelength_nm": 102.5175431905461,
      "density_harmonic_present": true,
      "z_matrix_element_a0": 0.0,
      "equal_three_state_z_amplitude_a0": 0.0,
      "electric_dipole_harmonic_present": false,
      "e1_angular_rule": "Delta l = +/-1 and Delta m = 0 for z polarization"
    },
    {
      "pair": "2p_z<->3s",
      "energy_gap_hartree": 0.06944444444444445,
      "frequency_hz": 456922494479166.5,
      "angular_frequency_rad_s": 2870928703831344.5,
      "vacuum_wavelength_nm": 656.1122764194948,
      "density_harmonic_present": true,
      "z_matrix_element_a0": 0.5417879392677545,
      "equal_three_state_z_amplitude_a0": 0.3611919595118363,
      "electric_dipole_harmonic_present": true,
      "e1_angular_rule": "Delta l = +/-1 and Delta m = 0 for z polarization"
    }
  ],
  "finite_grid_checks": {
    "audit_grid_shape": [
      81,
      81,
      81
    ],
    "audit_extent_each_axis_a0": [
      -14.0,
      14.0
    ],
    "phase_zero_norm": 1.0010695467306652,
    "analytic_continuity_max_abs": 0.0,
    "z_matrix_relative_error": 3.6982749751300888e-06,
    "integrated_current_relative_error": 0.0005343327450741566
  },
  "classification": {
    "equations": "written standard quantum mechanics",
    "matrices_and_rates": "finite deterministic computation",
    "visualization": "model output, not a measured electron movie",
    "measured_evidence": "none newly collected",
    "extra_dimensions": "not inferred or implemented"
  },
  "limitations": [
    "The current is probability flow, not a localized electron trajectory.",
    "The coherent packet is prepared, not the stationary hydrogen ground state.",
    "The spatial engine remains fixed-proton, non-relativistic, field-free, and spinless.",
    "The spontaneous rate is an ideal electric-dipole result, not a precision-QED lifetime calculation.",
    "A density beat alone does not identify an emitted-light harmonic; the observable matrix element is required."
  ]
}