On the uncertainty relation for Lz and φ

Loucas C. Papaloucas

Eric D. Chisolm

The proof of the Heisenberg uncertainty relation is modified to produce two improvements: (a) The resulting inequality is stronger because it includes the covariance between the two observables, and (b) the proof lifts certain restrictions on the state to which the relation is applied, increasing its generality. The restrictions necessary for the standard inequality to apply are not widely known, and they are discussed in detail. The classical analog of the Heisenberg relation is also derived, and the two are compared. Finally, the modified relation is used to address the apparent paradox that eigenfunctions of the z component of angular momentum Lz do not satisfy the φ–Lz Heisenberg relation; the resolution is that the restrictions mentioned above make the usual inequality inapplicable to these states. The modified relation does apply, however, and it is shown to be consistent with explicit calculations.

Tiago Pereira, Domingos H. U. Marchetti

We provide necessary and sufficient conditions for states to have an arbitrarily small uncertainty product of the azimuthal angle φ and its canonical moment Lz. We illustrate our results with analytical examples.

V. V. Ul'yanov, E. V. Ezerskaya

K. Kraus

ALFREDO LUIS ACOSTA PADILLA

We derive the orbital angular momentum operator Lz=mℏL_z = m\hbar Lz=mℏ and the orbital quantum number lmax⁡=3l_{\max}=3 lmax=3 for the electron entirely within the Wexp⁡/TW_{\exp}/T Wexp/T framework over Q(e)\mathbb{Q}(e) Q(e), without postulates from quantum mechanics. The electron is the trefoil knot T(2,3)T(2,3) T(2,3) in the V23 vacuum network. The crossing number c=3c=3 c=3 of the trefoil, forced by the degree of the Acosta polynomial x3−x−1=0x^3-x-1=0 x3−x−1=0, equals lmax⁡l_{\max} lmax. The angular momentum operator Lz=−iℏδϕL_z = -i\hbar\delta_\phi Lz=−iℏδϕ acting on eigenstates E(e,imϕ)E(e,im\phi) E(e,imϕ) of Wexp⁡/TW_{\exp}/T Wexp/T produces exact eigenvalues mℏm\hbar mℏ with mm m integer by the coherence condition of the orbital field in V23. The standard Heisenberg uncertainty explanation for atomic stability is replaced by a topological argument: the trefoil is a fibered knot with Alexander polynomial Δe(t)=t2−t+1≠0\Delta_e(t)=t^2-t+1\neq 0 Δe(t)=t2−t+1=0, and the Viète condition ρ1ρ2ρ3=1\rho_1\rho_2\rho_3=1 ρ1ρ2ρ3=1 on the roots of x3−x−1=0x^3-x-1=0 x3−x−1=0 guarantees it cannot dissolve. The quantum of action ℏ\hbar ℏ is derived as the total action of one complete phase cycle of the V23 network: ℏV23=(1/J0)⋅ℏJ0=ℏ\hbar_{V23} = (1/J_0)\cdot\hbar J_0 = \hbar ℏV23=(1/J0)⋅ℏJ0=ℏ exactly, where J0=ln⁡ρ/232J_0 = \ln\rho/23^2 J0=lnρ/232 is the torsional dissipation constant. The 2l+1=72l+1=7 2l+1=7 orientations of orbital l=3l=3 l=3 follow algebraically from l=3l=3 l=3 with no additional assumptions. The complete Class A chain: x3−x−1=0→c=3→T(2,3)→l=3→7x^3-x-1=0 \to c=3 \to T(2,3) \to l=3 \to 7 x3−x−1=0→c=3→T(2,3)→l=3→7 orientations. Zero free parameters. No quantum mechanical postulates.