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Axioms

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

Marlina Slamet, Viraht Sahni

Published Oct 16, 2026• 0 citations
Publisher

Abstract

We prove the Wave Function Identity (WFI), a new symmetry constraint on the solutions of the Schrödinger-Pauli equation for the Coulombically interacting N = 3 electron 2-dimensional 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 quartet state. The satisfaction of the Pauli principle (PP) and the odd parity of this state are also proved. An exact closed-form 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 of the WFI for the N = 2 electron case, we propose ad hoc the 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.

DOI
10.3390/axioms15100776

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Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.

Routledge eBooks•May 11, 2018•0 citations
We prove the Wave Function Identity (WFI), a new symmetry constraint on the solutions of the Schrödinger-Pauli equation for the Coulombically interacting N = 3 electron 2-dimensional 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 quartet state. The satisfaction of the Pauli principle (PP) and the odd parity of this state are also proved. An exact closed-form 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 of the WFI for the N = 2 electron case, we propose ad hoc the 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.
It is experimentally established that there is no ground triplet state of the natural He atom. There is also no exact analytical solution to the Schrödinger equation corresponding to this state. For a two-dimensional two-electron ‘artificial atom’ or a semiconductor quantum dot in a magnetic field, as described by the Schrödinger–Pauli equation, we provide theoretical proof of the existence of a ground triplet state by deriving an exact analytical correlated wave function solution to the equation. The state exists in the Wigner high-electron-correlation regime. We further explain that the solution satisfies all requisite symmetry and electron coalescence constraints of a triplet state. Since, due to technological advances, such a Wigner crystal quantum dot can be created, we propose an experimental search for the theoretically predicted ground triplet-state spectral line. We note that there exists an analytical solution to the Schrödinger–Pauli equation for a ground singlet state in the Wigner regime for the same value of the magnetic field. The significance to quantum mechanics of the probable experimental observation of the ground triplet state for an ‘artificial atom’ is discussed.
The Schrödinger-Pauli (SP) theory of electrons in an electromagnetic field explicitly accounts for the electron spin moment. The many-electron theory is complemented via a new descriptive perspective viz. that of the individual electron via its equation of motion or ‘Quantal Newtonian’ first law. The law is in terms of ‘classical’ fields whose sources are quantum mechanical expectation values of Hermitian operators taken with respect to the system wave function. The law states that each electron experiences an external and an internal field, the sum of which vanish. The external field is the sum of the binding electrostatic and a Lorentz field. The internal field is a sum of fields representative respectively of electron correlations due to the Pauli principle and Coulomb repulsion; the electron density; kinetic effects; and an internal magnetic field. The energy can be expressed in integral virial form in terms of these fields. The law is elucidated by application to the 23S state of a quantum dot in a magnetic field. It is proved that the Hamiltonian is an exactly known and universal functional of the wave function. This generalizes the SP equation, and reveals that its eigenfunctions and eigenvalues may be determined self-consistently. A Quantal density functional theory (QDFT) of the SP system is developed whereby additional properties are determined. A physical interpretation of Spin-DFT based on the QDFT mapping is provided. Further generalizations of the present work to the temporal case, and relativistic Dirac theory, are proposed.
Abstract The Schrödinger–Pauli (SP) theory of electrons in an electromagnetic field explicitly accounts for the electron spin moment. The many‐electron theory is complemented via a new descriptive perspective namely, that of the individual electron via its equation of motion or “Quantal Newtonian” first law. The law is in terms of ‘classical’ fields whose sources are quantum mechanical expectation values of Hermitian operators taken with respect to the system wave function. The law states that each electron experiences an external and an internal field, the sum of which vanish. The external field is the sum of the binding electrostatic and a Lorentz field. The internal field is a sum of fields representative respectively of electron correlations due to the Pauli principle and Coulomb repulsion; the electron density; kinetic effects; and an internal magnetic field. The energy can be expressed in integral virial form in terms of these fields. The law is elucidated by application to the 23S state of a quantum dot in a magnetic field. It is proved that the Hamiltonian is an exactly known and universal functional of the wave function. This generalizes the SP equation, and reveals that its eigenfunctions and eigenvalues may be determined self‐consistently. A quantal density functional theory (QDFT) of the SP system is developed whereby additional properties are determined. A physical interpretation of Spin‐DFT based on the QDFT mapping is provided. Further generalizations of the present work to the temporal case, and relativistic Dirac theory, are proposed.
Exact-diagonalization calculations for N=3N=3N=3 electrons in anisotropic quantum dots, covering a broad range of confinement anisotropies and strength of interelectron repulsion, are presented for zero and low magnetic fields. The excitation spectra are analyzed as a function of the strength of the magnetic field and for increasing quantum-dot anisotropy. Analysis of the intrinsic structure of the many-body wave functions through spin-resolved two-point correlations reveals that electrons tend to localize forming Wigner molecules. For certain ranges of dot parameters (mainly at strong anisotropy), the Wigner molecules acquire a linear geometry, and the associated wave functions with a spin projection Sz=1∕2{S}_{z}=1∕2Sz​=1∕2 are similar to the representative class of strongly entangled states referred to as the WWW states. For other ranges of parameters (mainly at intermediate anisotropy), the Wigner molecules exhibit a more complex structure consisting of two mirror isosceles triangles. This latter structure can be viewed as an embryonic unit of a zigzag Wigner crystal in quantum wires. The degree of entanglement in three-electron quantum dots can be quantified through the use of the von Neumann entropy.
For pt.II see Proc. R. Soc. London, Ser. A, vol.384, no.1786, p.89-105 (1982). A procedure for solving the few-particle Schrodinger equation exactly is applied to a model system consisting of two identical particles and a massive third particle. The type of interaction potential is not specified except that it should not diverge more rapidly than r -2 at the particle positions. Allowable interactions include the Coulomb and the harmonic oscillator potentials. The principles are illustrated by reference to the spatially symmetric states of the system. The solution has the form of a multipole expansion in spherical polar coordinates. The radial dependence for each multipole component is defined by a power series including logarithmic terms. Explicit expressions are given for the coefficients in the expansion. Exact coefficients are obtained for low-order terms in the wavefunction of a two-electron atom.
Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.