This paper is about a "number [which] has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it." [R. P. Feynman, QED: The strange theory of light and matter, Princeton 1985]. This number (Sommerfeld's fine-structure constant α = e 2 / hbar c ∼ 1/137) appears quite unexpectedly in the nonlinear mean-field Schrödinger-Poisson (SP) description of a non-relativistic quantum system; namely, quantum-dot helium constituted by a pair of 2D parabolically-confined oppositespin electrons entangled in the same singlet state. We show by iteration provided by a quickly convergent series of SP nonlinear eigenstates that the non-orthogonality (or inner product) of the two lowest "s" orbitals approaches asymptotically, within less than 0.5% error, the QED (quantum-electrodynamics) electronic amplitude e^iπ √ α for photon emission or absorption. The appearance of α in the non-relativistic SP system is explained by the role of virtual photons generated by the nonlinearity of the system.