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Re: [Phys-L] matter waves



For what it's worth, I have summarized the standard relations here:
https://www.usna.edu/Users/physics/mungan/_files/documents/Scholarship/MatterWaves.pdf

On Tue, May 16, 2017 at 7:48 PM, Moses Fayngold via Phys-l <
phys-l@mail.phys-l.org> wrote:


On Friday, May 5, 2017 12:57 PM, Carl Mungan <mungan@usna.edu> wrote:

>There is some discussion in various sources (such as Griffiths Intro to
Quantum Mech) about the phase velocity >of matter waves (say a beam of
monoenergetic nonrelativistic electrons). Some people find that it is
smaller >than the particle velocity, others that it is superluminal.

The phase velocity is uniquelydefined as u=omega/k=E/p, so it is
theproportionality coefficient between the frequency and propagation number
or between the energy and momentum. Therefore it can be measured by
measuring frequency andwave length. Measuring the interference pattern on
the screen in the doubleslit or diffraction grating experiment, we
indirectly measure the particle'sphase velocity. Equivalent result is
obtained in measuring the particle'senergy and momentum. If we know
particle's invariant mass and measure itsmomentum p, then its phase
velocity isfound from u = E/p = sqrt(m^2c^4+p^2c^2) /p = c
sqrt(1+m^2c^2/p^2) (1) Fora
tardyon (m real), Eq. (1) gives: u = u(p) >
c (1a)Fora photon (m=0) we have
u = const = c (1b)For a tachyon (m
imaginary, p>mc ): u = u(p) < c (1c)
According to (1), (1a, b , c), the value of u is not the question of who
does calculations,but what physical object is considered.
>In contrast, the group velocity equals the particle velocity.
It is not that simple.According to its definition v = d omega/dk = dE/dp,
the group velocity is a function of p. In other words, it is a local
characteristic v=v(p) in the momentum space– it generally depends on
position of a narrow sub-band you select within amomentum range of a wave
packet. In this respect, it is defined as uniquely asthe phase velocity.
One can reasonably argue that it represents the particle'svelocity V, but
that would hold onlyfor a packet very narrow in the momentum space.
Generally, the group velocityis not the particle's velocity. There is no
consensus about definition of theparticle's velocity V. And the reasonis
that the general concept of particle'svelocity has, unlike the phase
velocity, lost its significance in QM. Strictlyspeaking, it is an integral
characteristic – an expectation value over allpossible momenta in a wave
packet, like the packet's net momentum P itself: P = Int G(p)p dp
, V = Int G(p)(p/m gamma (v)) dp , (2)
where G(p) is Fourier-spectrum and gamma (v) is the Lorentz-factor.But in
some texts it is defined as the velocity of packet's maximum in
thecoordinate space, in some others – as the group velocity for the maximum
of themomentum spectrum. Generally, both are wrong – they differ from the
actually observedV as given in (2). This becomesevident for a packet with
more than one maximum.
My question is: Do you think there’s any physical significance to the
phase velocity of a matter wave? (For >example, to be practical, could some
experiment--say setting up a standing wave in a Bose-Einstein
condensate--reveal its value?)

I think, yes. Observingstanding wave is one possibility, and there are
others, e.g., mentioned in thebeginning of my comments.
If in contrast phase velocity is just a mathematical fiction, then I
guess there’s little point in arguing about why it >has different values
depending on how you calculate it.
According to (1), the phasevelocity, even though unsuitable for signaling,
is an important physicalcharacteristic of an object, and it has one value
for each state.

Moses Fayngold,NJIT


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Carl E. Mungan, Professor of Physics 410-293-6680 (O) -3729 (F)
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