◂ every box has its star

Robertson 1934 in every basis

Six observables, the pair (H, X) of each axis of the open tight binding cube, in its maximal current state. Recombine them with any real matrix M: Σ → M Σ Mᵀ and Ω → M Ω Mᵀ, so both determinants are multiplied by (det M)² and the ratio det Σ / |det Ω| does not move. The particle (its spacing a and hopping t) sets the scale through |det Ω|; the box sets the ratio, (Cx Cy Cz)², which lies in [((99 − 42√5)/5)³, (π²/3 − 2)³) for every box from 4 × 4 × 4 on.

The ratio does not move

Each new basis M changes det Σ and |det Ω| by many orders of magnitude (log scale). Their ratio stays on a flat line: (Cx Cy Cz)².

det Σ–
|det Ω|–
(det M)²–
det Σ / |det Ω|–
(Cx Cy Cz)²–
Cx, Cy, Cz–

Σ and Ω in the basis M

The 6 × 6 covariance matrix and bracket matrix after the change of basis. In M = 1 both are block diagonal, one block per axis; in a random basis every entry mixes.

The band of the cube

The ratio for this box (●) and for every box from 4 × 4 × 4 up to 25 × 25 × 25 (ticks). The floor is the regular 4 × 4 × 4; the ceiling (π²/3 − 2)³ is never reached. Growing the box moves the ratio up, away from saturation (ratio 1).

In Lean 4 (Physlib)

robertson_detRobertson 1934 for any C⋆-algebra: |det Ω| ≤ det Σ. Merged.
robertson_det_ratio_invariantdet Σ_b · |det Ω_a| = det Σ_a · |det Ω_b| for every square real M, any C⋆-algebra. Branch.
robertson_det_recombine_maxCurrentCubeState_strictOn the cube, from 4 × 4 × 4 on, |det Ω| < det Σ for every invertible M. Branch.
robertson_det_recombine_maxCurrentCubeState_mem_band((99 − 42√5)/5)³ |det Ω| ≤ det Σ < (π²/3 − 2)³ |det Ω| for every box and every invertible M.

The numbers on this page are computed in your browser from the matrices themselves: the state, H and X of each axis, their variances and brackets, then M Σ Mᵀ, M Ω Mᵀ and their determinants.

Built on Physlib's ProbabilisticTheory framework (T. O. Diem) and on robertson_det in Uncertainty.lean (T. O. Diem, J. Tooby-Smith, E. Nava-Hernandez).