Is phase velocity $c^2/v$ or $v/2$? Is there only one phase velocity as indicated by the above formula? If yes then why is wavepacket said to be superposition of 'waves' whose velocity is termed as phase velocity. Kindly correct me if my concepts are wrong as I am trying to learn quantum mechanics on my own.
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1please, edit the question. In its current state, its impossible to understand what you are asking. "why is it said..." who says this? "...waves whose velocity is termed as phase velocity" what does this even mean? – AccidentalFourierTransform Jan 12 '16 at 16:18
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Comment to the post (v2): It seems OP is asking about phase velocity and group velocity of matter waves, cf. e.g. http://physics.stackexchange.com/a/34215/2451 – Qmechanic Jan 12 '16 at 16:42
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I am also asking about the interpretation of wave packets – user103643 Jan 12 '16 at 17:21
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2Hi user103643. Welcome to Phys.SE. Please ask only one question per post. Please edit your post accordingly. – Qmechanic Jan 12 '16 at 18:24
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If you take the two well-known formulae 1. E = hν (Einstein) 2. pλ = h (de Broglie) and write the usual formula for phase velocity (v'), and do a bit of algebra, you end up with v' = E/p Now p is the momentum, which is well known to be = mv, v the particle velocity, m the mass, but what is E?
If you woodenly put in E = mc^2, you end up with v' = c^2/v, which is in your question. I object to that because while the Einstein formula is technically correct, the energy is unavailable while the particle stays in existence. Heisenberg put E = mv^2/2, i.e. he put E equal to the kinetic energy, but now we have the rather odd result that the wave travels at half the velocity of the particle. So, what you make of this is up to you. Most people ignore this relationship.
If you are only learning, you probably don't want the following. I have my own interpretation, which I call the guidance wave. It is like the pilot wave, but in the above I argue that if there is a wave that causes diffraction in the two-slit experiment, it has to travel at the same velocity as the particle. If so, the energy has to be mv^2, or twice the kinetic energy. That, in turn, means there is energy in the wave, or put it another way, the wave transmits energy (which is what every other wave does.) That has the ugly consequence of requiring effectively at least one more dimension in which this energy resides.
So, you see, there are three options for the phase velocity, and each has their problems. Take your pick. If you are studying for exams, I would strongly recommend you put this problem to one side. If you are trying to understand nature, then in the long term you don't really have that option, but in the short term, I would still put it to one side.
Ian Miller
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