Observables¶

All observables take a wavefunction (1D, 2D or 3D unless noted) and return plain floats or arrays.

Expectation values¶

Function	Quantity
`position_expectation`, `y_expectation`, `z_expectation`	⟨x⟩, ⟨y⟩, ⟨z⟩
`position_uncertainty`, `y_uncertainty`, `z_uncertainty`	Δx, Δy, Δz
`momentum_expectation(wf, direction)`	⟨p_x⟩, ⟨p_y⟩, ⟨p_z⟩
`momentum_uncertainty(wf, direction)`	Δp along an axis
`kinetic_energy_expectation`, `potential_energy_expectation`, `total_energy_expectation`	⟨T⟩, ⟨V⟩, ⟨H⟩
`transmission_coefficient`, `reflection_coefficient`	probability beyond / before a position along x

Momentum-space quantities use the Fourier transform of ψ, so they are exact for band-limited states. For a Gaussian packet Δx·Δp = ℏ/2 holds to 10⁻¹².

Currents, angular momentum and vortices¶

Function	Quantity
`probability_current(wf)`	j = (ℏ/m) Im(ψ*∇ψ) per axis
`gradient(psi, grid)`	spectral ∇ψ
`angular_momentum_z_expectation(wf)`	⟨L_z⟩ (2D/3D)
`vortex_positions(wf)`	positions and charges of quantised vortices in a 2D state, from the phase winding around every grid cell
`autocorrelation(psi0, psi_t, grid)`	C(t) = ⟨ψ(0)\|ψ(t)⟩. Its Fourier transform peaks at the eigenenergies (Feit–Fleck spectral method).

Gross–Pitaevskii quantities¶

Function	Quantity
`interaction_energy(wf, g)`	E_int = (g/2)∫\|ψ\|⁴
`gross_pitaevskii_energy(wf, V, g)`	the energy functional ⟨T⟩ + ⟨V⟩ + E_int, conserved by real-time GP dynamics
`chemical_potential(wf, V, g)`	μ = ⟨T⟩ + ⟨V⟩ + 2E_int

Pass potential=None for free space.