Abstract

When Schrödinger solved the Schrödinger equation for the hydrogen atom, he assumed the single-valuedness of the electronic wave function. Thereafter, this assumption has been one of the fundamental postulates of quantum mechanics. When wave functions are multi-component, however, the imposing of the single-valued condition may become nontrivial.The spin-degree-of-freedom of electron makes electronic wave functions two-component. When spin-vortices are created by the conduction electrons and they move in the self-consistent field with the spin-vortices, the twisting of the spin basis occurs; then, the imposing of the single-valued condition becomes nontrivial, and a vector potential is induced. As a consequence, the effective vector potential becomes the sum of the vector potential from the induced one and that originates from the electric current. This effective vector potential is gauge invariant and the persistent current is generated by it.In the present work, we argue that if interactions that are omitted in the BCS reduced Hamiltonian are included, spin-vortices may be generated upon the application of a magnetic field. Then, the vector potential is induced and provides with the phase variable, θ, of the electron pair amplitude. The appearance of the spin-vortex provides with a new origin of θ; it originates from the induced gauge potential. This origin is compatible with the superselection rule for charge in contrast to the currently-accepted origin.

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