Questions about deBroglie (matter) waves

[unusualname]

While I can follow the maths in the Dannon paper better than the Wagener paper, the follow quote taken from the latter does seem to raise some serious doubts about some of deBroglie’s assumptions.

“MacKinnon further points out that de Broglie emphasized the frequency associated with an electron, rather than the wavelength. His wavelength-momentum relationship occurs only once in the thesis, and then only as an approximate expression for the length of the stationary phase waves characterizing a gas in equilibrium. Most of MacKinnon’s article is devoted to analyzing the reasons why de Broglie’s formula proved successful, despite the underlying conceptual confusion. He finally expresses amazement that this confusion could apparently have gone unnoticed for fifty years.”​

While my present knowledge of the historical timeline of developments is only second-hand from reading a few books, it seems that de Broglie’s initial idea was not so much about describing particles in terms of a wave, but rather in terms of a particle having an associated ‘pilot wave’ that helped guide the particle through space and time. In this context, the ‘pilot wave’ theory was the first known example of a hidden variable theory, which was presented by Louis de Broglie in 1927. While this is possibly not within the scope of this thread, I would appreciate any pointers towards a description of the historical sequence of events. Thanks

There is a book (553 pages) freely available ar arXiv which has detailed discussion of de Broglie's pilot wave argument presented at the 1927 Solvay conference

Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference

 

[kith]

(Since κ and ω are both constant, it would not make sense to find the change, dω with respect to a change dκ.)

k and ω are constant but not independent. E = ℏω and p = ℏk, so the energy-impulse relation E(p) is equivalent to a frequency-wavenumber relation ω(k).

Group velocity for matter waves is also discussed in the wikipedia article:
http://en.wikipedia.org/wiki/Group_velocity

 

[mysearch]

There is a book (553 pages) freely available ar arXiv which has detailed discussion of de Broglie's pilot wave argument presented at the 1927 Solvay conference Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference

Many thanks, chapter 2 look to be very useful and readable.

 

[JDoolin]

k and ω are constant but not independent. E = ℏω and p = ℏk, so the energy-impulse relation E(p) is equivalent to a frequency-wavenumber relation ω(k).

Thank you. Right. They are constants in the equations as given, but both are functions of the particle-velocity.

$$\begin{matrix} \omega = \frac{E}{\hbar}=\frac{\gamma m c^2}{\hbar}=\frac{m c^2}{\hbar}\cosh(\varphi)\\ \kappa = \frac{p}{\hbar}=\frac{\beta \gamma m c}{\hbar}=\frac{m c}{\hbar}\sinh(\varphi)\\ \frac{\mathrm{d \omega} }{\mathrm{d} \kappa}=c \cdot \frac{\mathrm{d} (\cosh(\varphi))}{\mathrm{d} (\sinh(\varphi))} = c \tanh(\varphi)=c \beta =v_{particle} \end{matrix}$$

It turns out that the group velocity, as defined, does lead to the particle velocity, when using the formula for the traveling plane wave. (Of course, this wouldn't work if we used the nonrelativistic approximation, γ≈1)

 

[PhilDSP]

Hi,
Thanks for the reference in post #23, which I have obtained, although there seems to be a problem with the link, http://www.ptep-online.com/index_files/2010/PP-20-04.PDF" , which may be of interest to this discussion. Both this paper and the Wagener/MacKinnon papers appear to be raising some issues with the consistency of deBroglie’s logic in respect to wave mechanics.

The paper in the second link you provided doesn't appear to be a peer reviewed paper and neither affliliated with a recognized publisher. I don't believe it's quality is suitable for discussion here and any conclusions it might provide are highly suspect. Stating personal opinion: MacKinnon misses the single most important point of de Broglie theory but scores a dozen hits on detailed points. The paper at the second link misses pretty much everything even though the author exhibits a crude grasp of some issues. The full ramifications of de Broglie theory seem to be both subtle and paradigm shattering. The MacKinnon paper is also a nice detailed glimpse of the history and motivation of M. de Broglie and his theoretical ideas.

 

[PhilDSP]

There is a book (553 pages) freely available ar arXiv which has detailed discussion of de Broglie's pilot wave argument presented at the 1927 Solvay conference

Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference

Thanks so much. That appears to be an amazing piece of work! (Not to be assimilated any time soon though due to its enormous size...)

 

[KWillets]

$$\frac{\mathrm{d \omega} }{\mathrm{d} \kappa}=c \cdot \frac{\mathrm{d} (\cosh(\varphi))}{\mathrm{d} (\sinh(\varphi))} = c \tanh(\varphi)=c \beta =v_{particle}$$

Thanks for this -- I was thinking this morning it should be a nice short derivation.

 

[JDoolin]

Thanks for this -- I was thinking this morning it should be a nice short derivation.

You're welcome. I think hyperbolic trig is even cooler than regular trig. (Mostly because so few people know about it.)