The dynamics of superconductivity
The Meissner effect
and the flux quantization are two confirmations of
our general ideas. Just for the sake of completeness I would like to
show you what the complete equations of a superconducting fluid would
be from this point of view—it is rather interesting. Up to this
point I have only put the expression for $\psi$ into equations for
charge density and current. If I put it into the complete
Schrödinger equation I get equations for $\rho$ and $\theta$. It
should be interesting to see what develops, because here we have a
“fluid” of electron pairs with a charge density $\rho$ and a
mysterious $\theta$—we can try to see what kind of equations we get
for such a “fluid”!  So we substitute the wave function of
Eq. (21.17) into the Schrödinger
equation (21.3) and remember that $\rho$ and $\theta$ are
real functions of $x$, $y$, $z$, and $t$. If we separate real and
imaginary parts we obtain then two equations. To write them in a shorter
form I will—following Eq. (21.19)—write
\begin{equation}
\label{Eq:III:21:31}
\frac{\hbar}{m}\,\FLPgrad{\theta}-\frac{q}{m}\,\FLPA=\FLPv.
\end{equation}

One of the equations I get is then
\begin{equation}
\label{Eq:III:21:32}
\ddp{\rho}{t}=-\FLPdiv{\rho\FLPv}.
\end{equation}

Since $\rho\FLPv$ is first $\FLPJ$, this is just the continuity
equation once more. The other equation I obtain tells how $\theta$
varies; it is
\begin{equation}
\label{Eq:III:21:33}
\hbar\,\ddp{\theta}{t}=-\frac{m}{2}\,v^2-q\phi+
\frac{\hbar^2}{2m}\biggl\{
\frac{1}{\sqrt{\rho}}\,\nabla^2(\sqrt{\rho})\biggr\}.
\end{equation}

Those who are thoroughly familiar with hydrodynamics (of which I’m
sure few of you are) will recognize this as the equation of motion for
an electrically charged fluid if we identify $\hbar\theta$ as the
“velocity potential”—except that the last term, which should be
the energy of compression of the fluid, has a rather strange
dependence on the density $\rho$. In any case, the equation says that
the rate of change of the quantity $\hbar\theta$ is given by a kinetic
energy term, $-\tfrac{1}{2}mv^2$, plus a potential energy term,
$-q\phi$, with an additional term, containing the factor $\hbar^2$,
which we could call a “quantum mechanical energy.” We have seen that
inside a superconductor $\rho$ is kept very uniform by the
electrostatic forces, so this term can almost certainly be neglected
in every practical application provided we have only one
superconducting region. If we have a boundary between two
superconductors (or other circumstances in which the value of $\rho$
may change rapidly) this term can become important. For those who are not so familiar with the equations of hydrodynamics,
I can rewrite Eq. (21.33) in a form that makes the
physics more apparent by using Eq. (21.31) to express
$\theta$ in terms of $\FLPv$. Taking the gradient of the whole of
Eq. (21.33) and expressing $\FLPgrad{\theta}$ in terms
of $\FLPA$ and $\FLPv$ by using (21.31), I get

\begin{equation}
\label{Eq:III:21:34}
\ddp{\FLPv}{t}=\frac{q}{m}\biggl(-\FLPgrad{\phi}-\ddp{\FLPA}{t}\biggr)-
\FLPv\times(\FLPcurl{\FLPv})-(\FLPv\cdot\FLPnabla)\FLPv+
\FLPgrad{\frac{\hbar^2}{2m^2}
\biggl(\frac{1}{\sqrt{\rho}}\,\nabla^2\sqrt{\rho}\biggr)}.
\end{equation}


\begin{align}
\ddp{\FLPv}{t}&=\frac{q}{m}\biggl(-\FLPgrad{\phi}-\ddp{\FLPA}{t}\biggr)
-\FLPv\times(\FLPcurl{\FLPv})\notag\\[1ex]
\label{Eq:III:21:34}
&-(\FLPv\cdot\FLPnabla)\FLPv +
\FLPgrad{\frac{\hbar^2}{2m^2}
\biggl(\frac{1}{\sqrt{\rho}}\,\nabla^2\sqrt{\rho}\biggr)}.
\end{align}

What does this equation mean? First, remember that
\begin{equation}
\label{Eq:III:21:35}
-\FLPgrad{\phi}-\ddp{\FLPA}{t}=\FLPE.
\end{equation}

Next, notice that if I take the curl of Eq. (21.31), I
get
\begin{equation}
\label{Eq:III:21:36}
\FLPcurl{\FLPv}=-\frac{q}{m}\,\FLPcurl{\FLPA},
\end{equation}

since the curl of a gradient is always zero. But $\FLPcurl{\FLPA}$ is
the magnetic field $\FLPB$, so the first two terms can be written as
\begin{equation*}
\frac{q}{m}(\FLPE+\FLPv\times\FLPB).
\end{equation*}

Finally, you should understand that $\ddpl{\FLPv}{t}$ stands for the
rate of change of the velocity of the fluid at a point. If you
concentrate on a particular particle, its acceleration is the
total derivative of $\FLPv$ (or, as it is sometimes called in
fluid dynamics, the “comoving acceleration”), which is related
to $\ddpl{\FLPv}{t}$ by18
\begin{equation}
\label{Eq:III:21:37}
\left.\ddt{\FLPv}{t}\right|_{\text{comoving}}\kern{-2ex}=
\ddp{\FLPv}{t}+(\FLPv\cdot\FLPnabla)\FLPv.
\end{equation}

This extra term also appears as the third term on the right side of
Eq. (21.34). Taking it to the left side, I can write
Eq. (21.34) in the following way:
\begin{equation}
\label{Eq:III:21:38}
\left.m\ddt{\FLPv}{t}\right|_{\text{comoving}}\kern{-3.5ex}=
q(\FLPE\!+\!\FLPv\!\times\!\FLPB)\!+\!\FLPgrad{\frac{\hbar^2}{2m}
\!\biggl(\!\frac{1}{\sqrt{\rho}}\nabla^2\!\!\sqrt{\rho}\!\biggr)}.
\end{equation}

We also have from Eq. (21.36) that
\begin{equation}
\label{Eq:III:21:39}
\FLPcurl{\FLPv}=-\frac{q}{m}\,\FLPB.
\end{equation}
 These two equations are the equations of motion of the superconducting
electron fluid. The first equation is just Newton’s law for a charged
fluid in an electromagnetic field. It says that the acceleration of
each particle of the fluid whose charge is $q$ comes from the ordinary
Lorentz force $q(\FLPE+\FLPv\times\FLPB)$ plus an additional force,
which is the gradient of some mystical quantum mechanical
potential—a force which is not very big except at the junction
between two superconductors. The second equation says that the fluid
is “ideal”—the curl of $\FLPv$ has zero divergence (the divergence
of $\FLPB$ is always zero). That means that the velocity can be
expressed in terms of velocity potential. Ordinarily one writes that
$\FLPcurl{\FLPv}=\FLPzero$ for an ideal fluid, but for an ideal
charged fluid in a magnetic field, this gets modified to
Eq. (21.39). So, Schrödinger’s equation for the electron pairs in a superconductor
gives us the equations of motion of an electrically charged ideal fluid.
Superconductivity is the same as the problem of the hydrodynamics of a
charged liquid. If you want to solve any problem about superconductors
you take these equations for the fluid [or the equivalent pair,
Eqs. (21.32) and (21.33)], and combine
them with Maxwell’s equations to get the fields. (The charges
and currents you use to get the fields must, of course, include the ones
from the superconductor as well as from the external sources.) Incidentally, I believe that Eq. (21.38) is not quite
correct, but ought to have an additional term involving the
density. This new term does not depend on quantum mechanics, but comes
from the ordinary energy associated with variations of density. Just
as in an ordinary fluid there should be a potential energy density
proportional to the square of the deviation of $\rho$ from $\rho_0$,
the undisturbed density (which is, here, also equal to the charge
density of the crystal lattice). Since there will be forces
proportional to the gradient of this energy, there should be another
term in Eq. (21.38) of the form:
$(\text{const})\,\FLPgrad{(\rho-\rho_0)^2}$. This term did not appear
from the analysis because it comes from the interactions between
particles, which I neglected in using an independent-particle
approximation. It is, however, just the force I referred to when I
made the qualitative statement that electrostatic forces would tend to
keep $\rho$ nearly constant inside a superconductor.