Two kinds of momentum
The equation for the current is rather interesting, and sometimes
causes a certain amount of worry. You would think the current would be
something like the density of particles times the velocity. The
density should be something like $\psi\psi\cconj$, which is o.k. And
each term in Eq. (21.12) looks like the typical form
for the average-value of the operator
\begin{equation}
\label{Eq:III:21:13}
\frac{\Pcalvecop-q\FLPA}{m},
\end{equation}

so maybe we should think of it as the velocity of flow. It looks as
though we have two suggestions for relations of velocity to momentum,
because we would also think that momentum divided by mass,
$\Pcalvecop/m$, should be a velocity. The two possibilities differ by
the vector potential. It happens that these two possibilities were also discovered in
classical physics, when it was found that momentum could be defined in
two ways.7 One of them is called “kinematic
momentum,” but for absolute clarity I will in this lecture call
it the “$mv$-momentum.”  This is the
momentum obtained by multiplying mass by velocity. The other is a more
mathematical, more abstract momentum, sometimes called the “dynamical
momentum,” which I’ll call “$p$-momentum.” The two possibilities are
\begin{equation}
\label{Eq:III:21:14}
\text{$mv$-momentum}=m\FLPv,
\end{equation}
\begin{equation}
\label{Eq:III:21:15}
\text{$p$-momentum}=m\FLPv + q\FLPA.
\end{equation}
It turns out that in quantum mechanics with magnetic fields it is the
$p$-momentum which is connected to the gradient operator $\Pcalvecop$,
so it follows that (21.13) is the operator of a velocity.
 I’d like to make a brief digression to show you what this is all
about—why there must be something like Eq. (21.15) in
the quantum mechanics. The wave function changes with time according
to the Schrödinger equation in Eq. (21.3). If I would
suddenly change the vector potential, the wave function wouldn’t
change at the first instant; only its rate of change changes. Now
think of what would happen in the following circumstance. Suppose I
have a long solenoid, in which I can produce a flux of magnetic field
($\FLPB$-field), as shown in Fig. 21–2. And there is a charged
particle sitting nearby. Suppose this flux nearly instantaneously
builds up from zero to something. I start with zero vector potential
and then I turn on a vector potential. That means that I produce
suddenly a circumferential vector potential $\FLPA$. You’ll remember
that the line integral of $\FLPA$ around a loop is the same as the flux
of $\FLPB$ through the loop.8 Now what
happens if I suddenly turn on a vector potential?  According to the
quantum mechanical equation the sudden change of $\FLPA$ does not make a
sudden change of $\psi$; the wave function is still the same. So the
gradient is also unchanged. But remember what happens electrically when I suddenly turn on a
flux. During the short time that the flux is rising, there’s an
electric field generated whose line integral is the rate of change of
the flux with time:
\begin{equation}
\FLPE=-\ddp{\FLPA}{t}.
\end{equation}

That electric field is enormous if the flux is changing rapidly, and
it gives a force on the particle. The force is the charge times the
electric field, and so during the build up of the flux the particle
obtains a total impulse (that is, a change in $m\FLPv$) equal
to $-q\FLPA$. In other words, if you suddenly turn on a vector potential
at a charge, this charge immediately picks up an $mv$-momentum equal
to $-q\FLPA$. But there is something that isn’t changed immediately
and that’s the difference between $m\FLPv$ and $-q\FLPA$. And so the
sum $\FLPp=m\FLPv+q\FLPA$ is something which is not changed when you
make a sudden change in the vector potential. This quantity $\FLPp$ is
what we have called the $p$-momentum and is of importance in classical
mechanics in the theory of dynamics, but it also has a direct
significance in quantum mechanics. It depends on the character of the
wave function, and it is the one to be identified with the operator
\begin{equation*}
\Pcalvecop=\frac{\hbar}{i}\,\FLPnabla.
\end{equation*}
