Wave Function Identity: A New Symmetry for 2-electron Systems in an Electromagnetic Field

Authors

Marlina Slamet, Sacred Heart University
Viraht Sahni, CUNY Brooklyn CollegeFollow

Document Type

Article

Publication Date

10-3-2021

Abstract

Stationary-state Schrödinger-Pauli theory is a description of electrons with a spin moment in an external electromagnetic field. For 2-electron systems as described by the Schrödinger-Pauli theory Hamiltonian with a symmetrical binding potential, we report a new symmetry operation of the electronic coordinates. The symmetry operation is such that it leads to the equality of the transformed wave function to the wave function. This equality is referred to as the Wave Function Identity. The symmetry operation is a two-step process: an interchange of the spatial coordinates of the electrons whilst keeping their spin moments unchanged, followed by an inversion. The Identity is valid for arbitrary structure of the binding potential, arbitrary electron interaction of the form w(|r − r′|), all bound electronic states, and arbitrary dimensionality. It is proved that the exact wave functions satisfy the Identity. On application of the permutation operation for fermions to the identity, it is shown that the parity of the singlet states is even and that of triplet states odd. As a consequence, it follows that at electron-electron coalescence, the singlet state wave functions satisfy the cusp coalescence constraint, and triplet state wave functions the node coalescence condition. Further, we show that the parity of the singlet state wave functions about all points of electron-electron coalescence is even, and that of the triplet state wave functions odd. The Wave Function Identity and the properties on parity, together with the Pauli principle, are then elucidated by application to the 2-dimensional 2-electron ‘artificial atoms’ or semiconductor quantum dots in a magnetic field in their first excited singlet 2¹S and triplet 2³S states. The Wave Function Identity and subsequent conclusions on parity are equally valid for the special cases in which the 2-electron bound system, in both the presence and absence of a magnetic field, are described by the corresponding Schrödinger theory for spinless electrons.