I reckon the previous question was primarily about phase velocity
in general, and only secondarily about matter waves in particular.
It is generally a mistake, pedagogically and otherwise, to take
a question too literally. I've made this mistake often enough
to know rather well what it looks like.
So let me suggest a modified question: What, if anything, is moving
at the phase velocity?
As for the first part of the question, yes, some things *can* move
at the phase velocity, although other things cannot. In general,
tangible things cannot ... but there are lots of intangible things
in this world.
The canonical analogy is the point of intersection between the blades
in a shear. You could use common scissors, where the blades rotate,
but it is simpler to talk about a guillotine shear, where one blade
is stationary and the other moves in a straight line. They meet at
a slight angle, as you can see here, if you know what to look for: https://www.youtube.com/watch?v=2Cqbl6dY7jM&t=60
The point of intersection moves very much faster than the blade itself.
Optionally you can connect this to special relativity, where the
top blade is moving vertically at 0.2 times the speed of light,
while the point of intersection moves left-to-right ten times
faster, i.e. twice the speed of light. This is 100% true and
consistent with the laws of physics.
However, you don't need to know about relativity in order to
talk about phase velocity.
In any case, the point of intersection is not a tangible thing.
You can play the same game with people doing "the wave" in a stadium.
The wave can move right-to-left with an enormous velocity, even though
the people are not moving right-to-left at all. https://www.av8n.com/physics/img48/doing-the-wave.jpg
You can play the same game with shadows. Choose a geometry where the
shadow is very much longer than the object that casts it. If you move
the object parallel to itself, the shadow moves very much faster.
This has the advantage that you can make graphs of the wavefront
positions as a function of time. This can be used to illustrate
the idea that if you are trying to send /information/, that is
not determined by the phase velocity.
Observe that the Moiré pattern is moving very much faster than the
circular grid. That's a computer animation, but you can exhibit the
same phenomenon by printing grids on transparent foils. Put them on
an overhead projector, and/or pass them out for hands-on use.
This is an example of an intangible thing with practical applications,
e.g. the Vernier scale on a caliper or other instrument. You have to
give Pierre Vernier major props for taking something that was a mere
curiosity and turning it into something practical.
=================
Here's another line of thought that may help people with phase velocity:
In skilled hands, dimensional analysis is a powerful tool. In unskilled
hands, it can be a convenient way of obtaining wrong answers quickly.
Note the contrast:
++ A tangible object has some velocity.
-- Something with a velocity is not necessarily a tangible object.
Two things that both have dimensions of velocity are not necessarily
the same thing. Lagrangians, torques, and Gibbs free enthalpies all
have dimensions of energy, but they're not all the same thing, or
even the same kind of thing.
This also reaches back to the discussion of /one kind of charge/. A
proton has a positive charge, but it does not define what we mean by
charge, because there are lots of charged things that aren't protons.
A proton is a tangible thing, but charge is abstract and intangible.
To make the same point again, you can have a pion that decays into a
muon which decays into an electron ... in which case the charge is
the same, even though the particles are different.
The point is, intangible does not mean useless. Some of the most
profoundly useful things in physics are abstract and intangible.
This idea is super-hard to get across. A lot of students show up with
the idea -- implicit or explicit -- that
abstract = useless
intangible = fictional
and no amount of talking will persuade them otherwise. AFAICT the only
hope is to let them learn some useful examples, and then spiral back
and point out that the concepts they are using are in fact very very
abstract. For example, I suppose it would be "possible" to formulate
the laws of electromagnetism in terms of the number of protons, the
number of electrons, et cetera -- but it would be a colossal blunder.
Life is very much simpler if the laws are formulated in terms of the
plain old charge. There is only one kind of electric charge.