On Cosine Square as the Origin of the Heisenberg Uncertainty Lower Bound*

We first distinguish between: (1) the measurement-motivated momentum p = mv, with the related identity λk ≡ 2π (for all harmonic motions, of the tradeoff between λ, for the precision of the position, and the momentum as measured by k, in R¹⁺³), and (2) the momentum operator P:= (h/2πi)(d/dx), operating in a probabilistic framework, leading to the Heisenberg lower bound (h/4π), and referring to the velocity of the helical motion, te^it ≡ t·(cost, sint) ≡ (cost, sint, t) ∈ R³, where t ≡ x ∈ [a,a+λ] is an eigenvalue of the position operator X with probability distribution (│ψ(x)│²: x ∈ [a,a+λ]). I.e., P(X(ψ(x))) = (h/2πi)(d/dx)xe^ix = (h/2πi)e^ix + (h/2π)xe^ix, or P = (h/2πi) + (h/2π)X, so that σ_Pσ_X = (h/2π)σ_X². Basically, the wave-particle duality employs (cosx,sinx) ∈ R² for the wave and compresses it into e^ix ∈ C, with the particle observable x∈R multiplied as xe^ix through the multiplication operator E, which, if followed by the differential operator D, leads to an application of the product rule in differentiation: DE = i·I + ED. Since D and E are related by the Fourier-Plancherel operator F in D = FEF^-1, any pair of variables corresponded as such have the interpretation of D(E). We then elaborate on the derivation of the lower bound (h/4π) from a standard text, with the summary σ_Pσ_X = (h/2π)σ_X² ≧ (h/4π). Since the periodicity of the Poynting vector leading the electromagnetic wave of the photon is π, i.e., cos²π = cos²2π = 1, a photon has 0.5 probability to appear over [π,2π) of energy h (as accumulated from [0,π)), and 0.5 probability to appear over [2π,3π) of energy 2h (as accumulated from [π,2π)); i.e., a photon’s positions have E(X) ± σ_X,min = [2π-√2/2, 2π+√2/2] (radians). In summary, the 3-D motion (cosx, sinx, x), when compressed into xe^ix, lends itself to P(X) = [P,X] + XP = h·cos²π + h·cos²2π = 2h.