**Rings and tunnel junctions: Quantum mechanics on a circle**

We show by standard quantum principles that two circuits, a small tunnel junction and a small metal loop with an electron, are related by a gauge transformation. We show further that this same transform prevents momentum eigen functions from having gauge invariant de Broglie wave lengths around a ring. Thus persistent current on a metal ring and the Coulomb blockade on a small tunnel junction seem to be the same dynamical theory based on discontinuous Bloch waves on the perimeter of a circle. This is historically an area of simple quantum circuits where the principle of gauge invariance has not been applied. It is remarkable that some of these circuits are of intense current interest as candidates for qubit circuits for quantum computers.

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