Proof of the Wave Function Identity, a New Symmetry Constraint for the Three-Electron Quantum Dot in a Magnetic Field and Ad Hoc Generalization to Bound Many-Electron Systems

We prove the Wave Function Identity (WFI), a newly discovered property of solutions to the two-dimensional Schrödinger–Pauli equation for the N=3 electron Coulombically interacting harmonically bound `artificial atom’ or quantum dot in a uniform magnetic field. The symmetry is comprised of an interchange of the spatial coordinates of two electrons whilst keeping their spin moments unchanged, followed by an inversion of all three electron coordinates. The proof is achieved by first deriving the general form of the exact correlated wave function in the high-electron-correlation Wigner regime and then proving the WFI employing this form for a ground and quartet state. The satisfaction of the Pauli principle (PP), and the odd parity of the quartet state which follows from the WFI, are also proved. An exact analytical expression for a quartet state in the Wigner regime is derived, and the properties of the PP, WFI, parity and other properties exhibited diagrammatically. Together with our prior proof that the exact wave functions of the N=2 electron systems satisfy the WFI, we conjecture the ad hoc validity of the WFI for all bound-state N≥2 electron systems for arbitrary symmetric binding potential, arbitrary electron-interaction of the form ω(|r−r′|) and arbitrary dimensionality.