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A magnetic flux quantum is the flux scaleassociated with a coherent degree of freedom of electric charge $q\ne 0$. Its Aharonov-Bohm phase on a loop enclosing magnetic flux $\Phi$ is $q\Phi /\hslash$; adding ${\Phi }_0$ changes that phase by one full turn.

For an electron, the magnitude of the charge is $e$, giving $h/e$. A conventional superconducting Cooper pair has charge magnitude $2e$, giving $h/(2e)$. The relevant charge must therefore be stated: a superconducting flux scale and a single-electron interference period differ by a factor of two.

In the magnetic translation algebra, translations along two lattice vectors commute when the enclosed unit-cell magnetic flux obeysThis is a condition on the algebra for the specified cell, not a claim that every externally applied magnetic flux must be quantized. The distinct magnetic flux spectrum of a quantum ring illustrates flux-periodic energy levels.

For the magnetic flux spectrum of a quantum ring, the periodic boundary condition permits the normalized plane waves $L^{-1/2}{e}^{2\pi inx/L}$, with $n\in \mathbb{Z}$. Their energy eigenvalues areShifting $\alpha$ by an integer relabels $n$, so the ordered energy levels have period one in $\alpha$, or $2\pi \hslash /L$ in $a$. For $-1/2\le \alpha \le 1/2$, the lowest two areextended periodically. The lowest branches cross at every half-integer $\alpha$, with energy $E_L/4$. At integer $\alpha$, the first excited energy level has multiplicity two. The corresponding magnetic flux period is the electron magnetic flux quantum $h/e$.