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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.
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)
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?)
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.