In the Landau-Ginzburg model of superconductivity, the energy of the system is given, for constants $\alpha$ and $\beta$, by

$$E=\int\left\{\frac{1}{2 \mu_{0}} \mathbf{B}^{2}+\frac{1}{2 m}\left[(i \hbar \nabla-q \mathbf{A}) \psi^{*} \cdot(-i \hbar \nabla-q \mathbf{A}) \psi\right]+\alpha \psi^{*} \psi+\beta\left(\psi^{*} \psi\right)^{2}\right\} d^{3} \mathbf{x},$$

where $\mathbf{B}$ is the time-independent magnetic field derived from the vector potential $\mathbf{A}$, and $\psi$ is the wavefunction of the charge carriers, which have mass $m$ and charge $q$.

Describe the physical meaning of each of the terms in the integral.

Explain why in a superconductor one must choose $\alpha<0$ and $\beta>0$. Find an expression for the number density $n$ of the charge carriers in terms of $\alpha$ and $\beta$.

Show that the energy is invariant under the gauge transformations

$$\mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda, \quad \psi \rightarrow \psi e^{i q \Lambda / \hbar}$$

Assuming that the number density $n$ is uniform, show that, if $E$ is a minimum under variations of $\mathbf{A}$, then

$$\operatorname{curl} \mathbf{B}=-\frac{\mu_{0} q^{2} n}{m}\left(\mathbf{A}-\frac{\hbar}{q} \nabla \phi\right)$$

where $\phi=\arg \psi$.

Find a formula for $\nabla^{2} \mathbf{B}$ and use it to explain why there cannot be a magnetic field inside the bulk of a superconductor.