"Group velocity (equal to an electron's speed) should not be confused with phase velocity (equal to the product of the electron's frequency multiplied by its wavelength)." Above statement is from wikipedia, and I still don't quite understand it. As one of the postulate of QM, group velocity represent the velocity that a particle is moving. However, in DeBrogile relationship, after we know the speed of electron c, then we can know the wavelength and frequency by using wavelength=h/mc. and f=c/wavelength. Does the speed c here differs from the group velocity of electrons. I am quite new to quantum mechanics, and appreciate if anyone help to explain on this topic, thanks
Group velocity is considered the propagation speed of the wave (if the entity is a wave) or the velocity of the particle (if the entity is a particle) Phase velocity is typically much higher in value and is not considered to have direct physical significance. It tells you how fast the mathematical plane is moving which generates the harmonic fluccuations in the wave pattern. In a light wave in a pure vacuum the phase velocity equals the group velocity which also equals c - the speed of light. Matter and de Broglie waves associated with matter will have a lower group velocity but a higher than c phase velocity.
I have some doubts about your relation for the frequency of the electron. How would you determine experimentally that frequency?
Just remember that the group velocity is also not free from interpretational issues. For example, as has been demonstrated in fast light experiments, it's possible to create a material in which the group velocity for a photon is higher than c, and even negative. Negative group velocity would mean that, say, the peak of the wavepacket exits the material before it enters. This still does not violate causality, so the lesson here is that the group velocity of an object does not correspond to the speed by which the information from this object is sent.
On the wikipedia website there is a nice animation illustrating the difference between group and phase velocity: http://en.wikipedia.org/wiki/Group_velocity
It can't be measured directly of course. It's wavenumber and angular frequency are determined according to the de Broglie relations for the photon of a measured frequency. The phase velocity is then the angular frequency divided by the wavenumber.
In de Broglie's textbook "Introduction to the Study of Wave Mechanics" he analyzes the results of the first few experiments that were shown to prove his theory of "matter waves". Especially in G. P. Thomson's experiments, a large number of trials with different parameters were made each yielding a unique result. Those results would be very, very difficult to explain if anything other than a wave accompanying the electron were to produce them. He photographed the circular pattern of the wave of single electrons when shot through various ultra-thin wafers of material and could easily bend the pattern by applying a magnetic field.
"" Negative group velocity would mean that, say, the peak of the wavepacket exits the material before it enters."" It does not mean that. Waves build up at the front of the photon and move backwards through the photon until they fade away at the back of the photon. Positive phase velocity means that the waves build up at the back and move forward through the photon and fade away at the front. Neither can carry information faster than the photon.
Tbh, I don't quite get what you mean, though note that I made a comment about group velocity, not phase velocity. What I was refering to is the fact that by tailoring the group refracting index of a material in special ways, one can make the group velocity have any number, including such values that gives rise to speeds higher than c or even negative speeds. The meaning of a negative velocity really is that a particluar chosen point in a pulse packet, for example the peak of the pulse, will exit such a prepared material before the peak of the incoming pulse enters it. This behavior has been experimentally verified, for example here: L. J. Wang, A. Kuzmich and A. Dogariu. Gain-assisted superluminal light propagation. Nature 406, 277 (2000), where the pulse exited the material 62 ns (19 meters travelled) before it entered. The point is still clear however, that while the peak may travell faster than c, information transfer from such a pulse never exceeds c, because the peak of a wavepacket is not a good definition of the information content.
You are right .You did say group velocity. The faster than light propagation of a photon in the special material sounds like a trick of words. As the photon enters the material the peak is at the back?. As the front of the photon leaves the material the peak is at the front,a different wave (in the same photon) but now bigger (modified by the material) so it is now the photon peak. The back of the photon has not entered the material yet but the front is coming out. So the photon peak at the front leaves before (a different peak ) at the back enters. The photon must be longer than the thickness of the material. No problem with that. The same wave OUT before IN violates causality so cannot be. FTL is OK by me. Causality violations are not OK. How can such small time intervals be measured?. Was it all just a calculation?.