Faster than light in Quantum mechanics?

[stgdf01]

In <Engineering Electromagnetics> written by W.H.Hayt and J.A.Buck( 6th edition,McGraw-Hill,p372), the phase velocity v_p of electromagnetic waves in copper at 60Hz (commercial electric power) is 3.2m/s. Substituting the value into v_pv = c² of de Broglie theory, v=10⁸c should be much faster than light speed c in vacuum?

 

[PhilDSP]

Hi stgdf01 and welcome to the forum,

The analysis you posted sounds incorrect. Phase velocity (of the EM wave) must always be greater or equal to c. The analysis seems to have been done with the assumption that the AC voltage wave represents the actual movement of energy. It doesn't. The energy travels far quicker than the swinging of the voltage. But maybe the person doing the analysis meant the velocity of the conduction current? If so, then group velocity v_g rather than phase velocity is the correct term to use.

v_p v_g= c²

 

[stgdf01]

The analysis you posted sounds incorrect. Phase velocity (of the EM wave) must always be greater or equal to c. The analysis seems to have been done with the assumption that the AC voltage wave represents the actual movement of energy. It doesn't. The energy travels far quicker than the swinging of the voltage.

The study to v_p in conductors is based on first principle(Maxwell's equations) and has nothing to do with parameters in circuit theory such as voltage,inductance and capacitance. For example, J.D.Jackson's <Classical Electrodynamics> and D.J.Griffiths's <Introduction to Electrodynamics>.

If so, then group velocity v_g rather than phase velocity is the correct term to use.

v_p v_g= c²

The group velocity v_g=dw/dk of low-frequency electromagnetic field in conductors is twice as much as v_p=w/k (P.Lorrain, D.R.Corson, F.Lorrain, <Electromagnetic Fields and Waves>, 3rd edition). For commercial power in copper, v_p=3.2m/s and v_g=6.4m/s. In this case, neither v_p nor v_g can play the role of c²/v_p in de Broglie's theory. Actually, it is v=p/m (m=E/c²) in point mechanis.

 

[PhilDSP]

In this case, neither v_p nor v_g can play the role of c²/v_p in de Broglie's theory.

It sounds likely that the analysis involved integration over macroscopic areas of space and large increments of time. In that case the average variation of the EM values would have washed out the high frequency effects. That's what I was trying to express in the first response. The analysis apparently ignores quantum effects.

 

[Cthugha]

Substituting the value into v_pv = c² of de Broglie theory, v=10⁸c should be much faster than light speed c in vacuum?

This result of de Broglie theory is usually explicitly derived for matter waves. You will of course get strange results when you try to use it without proper modification for particles without rest mass and em waves.

 

[navilon]

neutrinos as being said faster than light,,have very small mass and could be compound of gravity also accelerating speed such as dark energy particles..therefore interacting with light much like gravity does..as a mediator..a boson wave..already there..QED

 

[stgdf01]

This result of de Broglie theory is usually explicitly derived for matter waves. You will of course get strange results when you try to use it without proper modification for particles without rest mass and em waves.

E=hw and p=hk is of course tenable to photons otherwise the Compton effect, photoelectric effect, blackbody radiation and others cannot be interpreted. In history, these two relations are concluded from experiments to massless photons and then de Broglie extended to massive matter waves(electrons,say).

 

[Cthugha]

E=hw and p=hk is of course tenable to photons[...]

Sure, but that does not have anything to do with v_p v=c^2. This result is derived explicitly in the nonrelativistivc limit for matter waves and requires w and p to depend on mass. Some good textbooks on this topic state this explicitly. Even the really basic about about basic knowledge for engineers by Hennecke mentions this.

 

[PhilDSP]

Sure, but that does not have anything to do with v_p v=c^2. This result is derived explicitly for matter waves and requires w and p to depend on mass. Almost every good textbook on this topic states this explicitly. Even the really basic about about basic knowledge for engineers by Hennecke mentions this.

Actually the history behind that for waves in general extends much further back in time:

The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.[6]

http://en.wikipedia.org/wiki/Group_velocity
http://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleigh

 

[stgdf01]

Sure, but that does not have anything to do with v_p v=c^2. This result is derived explicitly in the nonrelativistivc limit for matter waves and requires w and p to depend on mass. Some good textbooks on this topic state this explicitly. Even the really basic about about basic knowledge for engineers by Hennecke mentions this.

But in optics and electromagnetism, v_p v=c^2. is still valid to a field(photons) in vacuum. On the other hand, the photon can be described in relativistic mechanics as a special case of V=c and m_0=0

 

[stgdf01]

It sounds likely that the analysis involved integration over macroscopic areas of space and large increments of time. In that case the average variation of the EM values would have washed out the high frequency effects. That's what I was trying to express in the first response. The analysis apparently ignores quantum effects.

I regards this as a counter-example of v_p v_g=c^2,although v_p v=c^2 is universal.That is to say, the group velocity v_g is not always equal to V. It can be applied to explain why those experiments of superluminal group velocities v_g>c are not really faster than light v>c.

 

[Cthugha]

But in optics and electromagnetism, v_p v=c^2. is still valid to a field(photons) in vacuum.

Yes, in vacuum. You were interested in light in a material, no?

On the other hand, the photon can be described in relativistic mechanics as a special case of V=c and m_0=0

You can treat a photon as having m_0=0 if you include the full relativistic mass E=\sqrt{(m_0 c^2)^2 +(p c)^2} in the derivation.

However, you get the v in v_p v=c^2 in common de Broglie theory by using p=m v. For photons you do not have this explicit dependence and just use p=\frac{h}{\lambda}, so you only arrive at the trivial v_p =\frac{c}{n}.

 

[PhilDSP]

If we know or can determine the dispersion equation (which doesn't include a mass term) then we can find the phase velocity algebraically and the group velocity by taking the derivative of the dispersion equation (or its roots) with respect to wave number.

v_p = \frac {\omega}{k} \ \ \ \ \ \ v_g = \frac {\partial \omega}{\partial k}

For example: the FT of \ \ \ (\nabla^2 - \frac {1}{c^2} \frac {\partial^2}{\partial t^2}) \phi = 0 \ \ \ is \ \ \ (-k^2 + \frac {\omega^2}{c^2}) \phi = 0 \ \ \ so that \ \ \ D(\omega, k) = -k^2 + \frac {\omega^2}{c^2}

then \ \ \ \ \omega = ±ck \ \ \ \ giving \ \ \ \ v_p = ±c \ \ \ \ and \ \ \ \ v_g = ±c \ \ \ \