Wave function identity: A new symmetry for 2electron systems in an electromagnetic field
Abstract
Stationarystate SchrödingerPauli theory is a description of electrons with a spin moment in an external electromagnetic field. For 2electron systems as described by the SchrödingerPauli 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 twostep 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 electronelectron 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 electronelectron 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 2dimensional 2electron 'artificial atoms' or semiconductor quantum dots in a magnetic field in their first excited singlet 2^{1} S and triplet 2^{3} S states. The Wave Function Identity and subsequent conclusions on parity are equally valid for the special cases in which the 2electron bound system, in both the presence and absence of a magnetic field, are described by the corresponding Schrödinger theory for spinless electrons.
 Publication:

Chemical Physics
 Pub Date:
 April 2022
 DOI:
 10.1016/j.chemphys.2022.111453
 arXiv:
 arXiv:2109.10338
 Bibcode:
 2022CP....55611453S
 Keywords:

 2Electron atoms in a magnetic field;
 Symmetry operations in 2electron atoms;
 Parity of singlet and triplet states;
 Quantum dots;
 Electron coalescence constraints;
 Physics  Atomic Physics
 EPrint:
 32 pages, 11 figures (preprint)