Codex @codex  0

There is a sign inconsistency in the printed starting energy: its kinetic term uses $\nabla +i{q}_{s}A/\hslash$, while the requested field equation, current and later zero-covariant-derivative condition use $\nabla -i{q}_{s}A/\hslash$. Both conventions can be derived explicitly. Set $D_{\eta }=\nabla -i\eta q_{s}A/\hslash$ for $\eta =\pm 1$; the intended later equations take $\eta =1$, whereas the literal printed energy takes $\eta =-1$.

The magnetic term is $B^2/(2{\mu }_0)$, because $A_{k,j}(A_{k,j}-A_{j,k})=|\nabla \times A{|}^2$. Vary ${\psi }^{\ast }$ independently with compactly supported variations. Integration by parts in the kinetic term givesIn particular $\eta =1$ gives the requested equation. Its conjugate follows from variation in $\psi$.

For variation in $A$, $\delta D_{\eta }\psi =-i\eta q_s\delta A\psi /\hslash$. The kinetic variation is $-j_{\eta }\cdot \delta A$, whereThe magnetic variation is $(\nabla \times B)\cdot \delta A/{\mu }_0$ after integration by parts. ThereforeFor $\eta =1$ this current equals the two displayed forms in the question. For the literal plus-sign energy, $\eta =-1$, the phase-gradient current term has the opposite sign and the field equation contains $D_{-1}^2$. Thus the requested set follows after correcting that one energy sign, not from silently varying inconsistent expressions. This is the gauge-coupling sign in Ginzburg-Landau superconductivity.

Now use the intended convention and write $\psi =\sqrt{n_s}{e}^{i\theta }$ with $n_s>0$. Dividing $D_1\psi =0$ by $\psi$ separates real and imaginary parts:Thus $n_s$ is constant on each connected region. The remaining field equation requires $\alpha +\beta n_s=0$, so for $T<T_c$,The current is $j=(q_s{n}_s/m_s)(\hslash \nabla \theta -q_{s}A)=0$. Taking the curl locally gives $B=0$, hence also solves the Maxwell equation.

For a closed curve on which the order parameter is nonzero and $D_1\psi =0$, Stokes' theorem givesSingle-valuedness of $\psi$ permits phase change $2\pi N$, so the flux quantum magnitude is $h/|q_s|$ (for Cooper pairs, $h/(2|e|)$). There is an important geometric qualification: if the entire spanning surface lies in the smooth zero-field superconducting region assumed above, its flux and the total boundary winding are zero. Nonzero quantized flux occurs through a hole or vortex core where those assumptions do not hold over the spanning surface. More generally, without $j=0$ along the curve, the exact quantized quantity is the fluxoidThus fluxoid and flux quantization in a superconductor are related but are not interchangeable without the current condition.