# What is a rigid wave function?

The London equations prior to BCS that describe superconductivity require assuming the wavefunction describing the superconducting pair of electrons to be rigid. I've been looking all over trying to find what does it mean for a wave function to be rigid. All I have found is that the phase of the wave function is what is actually assumed to be rigid, and even then none of these sources explain what it means but rather just use it or assume the reader knows.

What does it mean for a wave function to be rigid? And why is this a necessary condition for the validity of the London equations?

As you might have seen the London brothers were the first to construct a successful theoretical model of superconductivity, translated in the two famous London equations, of which the second one, $$\textbf{B}=-\mu_0\lambda_L^2 \nabla\times\textbf{j}_s$$ describes the Meissner effect. Now this was more of a phenomenological theory, and I won't go into detail but the basic idea was that they assumed a spatially constant density of superconducting charge carriers (later we figured out that these were Cooper pairs).
However, later this phenomenological model was extended by Ginzburg and Landau. They allowed the variation of the Cooper pair density, which was assumed to be constant in London's theory. Furthermore, they introduced a characteristic length scale over which the Cooper density (order) can change nl. the Ginzburg-Landau coherence length $$\xi_{GL}$$. This was an important step forward because by introducing this coherence length $$\xi_{GL}$$ one can distinguish between type I and type II superconductors by the relative length scale of $$\xi_{GL}$$ w.r.t. $$\lambda_L$$.
Both of these superconductor types can be in the Meissner state below a certain critical temperature and magnetic field (type II can also be in the Shubnikov phase above between certain critical temperatures and fields). We expect the Meissner state to occur when we have a strong expulsion of the external magnetic field, which means that the coherence length $$\xi_{GL}$$ needs to be sufficiently large compared to the penetration depth $$\lambda_L$$. This will indeed be the condition for the Meissner state to occur. That is what they mean, in your text when they speak about the rigidity of the wave function (or alternatively they named it 'superconducting coherence') nl. a large coherence length; spatially homogenous Cooper density (see p.41 of link: 'second London equation can only be derived from BCS theory by assuming a spatially homogenous BCS state).
Note: I would recommend reading the standard text on superconductivity by Michael Tinkham. Because in the link you mentioned they are not very careful with the distinction between the different characteristic length scales. In fact, there are three main characteristic length scales: London penetration length $$\lambda_L$$, GL coherence length $$\xi_{GL}$$, and Cooper pair length $$\xi_0$$.