Re: [Phys-L] matter waves

[Moses Fayngold <moshfarlan@yahoo.com>]

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