De Broglie wavelength and mass

[unchained1978]

I've been studying quantum physics for a little over a semester now and I'm having trouble wrapping my head around the physical significance of matter waves. The derivation also confuses me because it states the momentum of particle with mass as mc, which is only true for a massless photon. I don't understand what is waving in this situation and also, how is the phase velocity of a matter wave always > c for a particle moving with subluminal velocity?
Any explanations or suggested reading would be appreciated.

 

[javierR]

As for the relations involved: The energy of a photon is E=pc. Then due to the relationship E=h*nu of Planck and Einstein, the momentum is p=h*nu/c. Finally because lambda*nu = c for a photon, p= h/lambda.
For a non-relativistic particle with mass m, E=p^2/2m. De Broglie suggested that the energy of an electron could also come in fixed amounts of energy E, and in the same way as for a single photon, it could be that it depends only on the frequency associated with the particle: E=h*nu. Using the fact that p=mv and lambda*nu= v for a wave propagting with speed v, we find that p=h/lambda holds for the massive particle too.
So E=h*nu and p=h/lambda are true for both photons and electrons, but the relation between the wavelength and frequencies are different: lambda*nu = speed.
The meaning of the wave-nature for a matter particle is the same as the wave-nature of a single photon. Both cases should therefore be equally as strange. Instead of picturing particles as little BB's, try picturing a wavepacket (a waveform with finite length) whose amount of spread in space is a measure of the amount of indefiniteness in position in space.
Technically, the waveforms are complex, having a real and imaginary part, but de Broglie didn't know about this yet...he suggested just a real waveform in space. In particular, his suggestion for the waveform closing into a circular orbit corresponds to saying the particle's position on the orbital path is completely indefinite. It doesn't have a single point position in space.

 

[PhilDSP]

To fully partake of the concept of Matter Wave you probably need to consider exactly what de Broglie proposed. He did not say there existed a wave or wave packet that can take on particle form but rather that the particle had an associated but separate wave. In simple terms the particle creates ripples in the background EM fields as it travels.

The always greater than c phase velocity for matter is a consequence of Maxwell's Equations when analyzed in terms of phase and group velocity. A traveling electron, for instance, causes a moving dispersion of all field lines. That dispersion decreases with increasing velocity - its "footprint" grows smaller. Another way to view it is that the proportion of conductance current gains on displacement current.

De Broglie actually did consider the phase of the waves (real and imaginary solutions of the equations) But apparently the big names in the field were skeptical of much of de Broglie's analysis and ideas at the time he presented them and didn't follow his line of thinking very far.