Questions about deBroglie (matter) waves

[KWillets]

My reference gives:

$$c_{phase}=\frac{\omega}{\kappa}$$

and

$$c_{group}=\frac{d\omega}{d\kappa}$$

where

$$\kappa=2\pi/\lambda$$

 

[mysearch]

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 Dannon paper seems to go further by questioning the interpretation of the deBroglie equation in terms of wavelength by linking this measure of distance to the uncertainty principle. While the maths seems logical, the symbols used for velocity and frequency can be confusing, but maybe somebody with a wider understanding of these issues might wish to comment further. However, I would like to initially highlight just one aspect of this paper, see the end of section 2, page 5, regarding the interpreted nature of ‘matter waves’. While the similarities between the Compton and deBroglie equations can be seen, Compton’s wavelength is constrained to photons, alias EM waves, where the propagation velocity is unambiguously defined by c. However, the deBroglie equation also seems rooted in equating the Planck and Einstein expression for energy plus a fundamental assumption about the relationship between velocity, frequency and wavelength.
$$mc^2=hf=h\frac {c}{\lambda}$$
I would also like to clarify a few points made in post #27:

………. it means the wavelength and frequency are not proportional, and are not related by a simple formula

While I did say that this was a good point, in the context of the discussion, this was not my position, only the inference that might be drawn following on from the deBroglie equation. Again, while I did post the following equation, it was only to confirm the deBroglie logic that appears to suggest a phase velocity >c.
$$v_p=f \lambda= \frac {E}{h} \frac {h}{p} = \frac {E}{p} = \frac {mc^2}{mv} = \frac {c^2}{v}$$
As has been raised several times, I have some concerns about the direct extrapolation of momentum in the Compton equation to the deBroglie equation, which seems to be implicit in the equation above.
$$\lambda = \frac {h}{mc} = \frac {h}{p}= \frac {h}{mv}$$
Finally, with reference to the wave equations in post #31, which I agree, but would like to generalise to a form where the velocity equals v, not c.
$$v_p=\frac {\omega}{\kappa}=f \lambda$$
It is my understanding that this equation expresses a fundamental ‘geometry’ of any ‘travelling wave’. I am using quotes because this terminology might not be semantically correct, although I hope the correct meaning is conveyed. In this context, a ‘group wave’ or a ‘standing wave’ appears not to be waves, but rather the product or superposition of 2 or more ‘travelling waves’, i.e. they represent an inference pattern, which can have a propagation ‘group’ velocity 0<v<c on the assumption that c is the highest physical velocity possible of any 'travelling wave'.
$$v_g=\frac{d\omega}{d\kappa}$$
Anyway, I would appreciate any insights regarding the Dannon paper. Thanks

 

[PhilDSP]

I took a peek at de Broglie's thesis paper again for the first time in a few years (my reference is his book "Introduction to the Study of Wave Mechanics"). I can see why MacKinnon believed de Broglie was wrong and confused. The whole small first chapter on "phase waves" is not at all helpful - probably downright misguiding... It was no doubt scaffolding that germinated the proper theory but deserved to be excised in a proper and mature exposition of his theory.

 

[mysearch]

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

 

[JDoolin]

De Broglie's original development, however, relied on an incorrect identification of two quite different relations: the relation between the velocity of a particle and the relation between the group velocity of a wave packet and the velocity of individual waves in the packet.

It seems to me there are three velocities involved here; particle velocity v, phase velocity v_p, and group velocity, v_g

I think it might be productive for me to go back and review the derivation of formulas for wave and group velocity, but for someone who has seen or taught these derivations, does this relationship v_p = \lambda f hold for both standing waves, plane waves, and everything in between, or is it a relationship which holds only for plane waves?

My reference gives:

$$c_{phase}=\frac{\omega}{\kappa}$$

and

$$c_{group}=\frac{d\omega}{d\kappa}$$

where

$$\kappa=2\pi/\lambda$$

Your answer suggests a relationship that may hold in general for a function y(x,t):

(1)
$$\begin{matrix} \frac{\partial y}{\partial t} = \omega y\\ \frac{\partial y}{\partial x} = \lambda y \end{matrix}$$

As an example:

A standing wave has an equation like:
$$y=sin(2 \pi \frac{x}{\lambda})sin(2 \pi \frac{t}{T})$$

While a traveling wave has an equation like:
$$y=sin(2 \pi (\frac{x}{\lambda}-\frac{t}{T}))$$

And a deBroglie wave has an equation somewhere in between?

Setting:
$$\omega = \frac{2 \Pi}{T} = 2 \Pi f$$
and

$$\kappa = \frac{2 \Pi}{\lambda}$$

The standing wave becomes (2)
$$y=sin( \kappa x )sin( \omega t )$$

while the traveling wave becomes (3)
$$y=\sin( \kappa x - \omega t)$$

The relationship (1) holds for both the standing wave (2) and the traveling wave (3).

However, if I use v_{phase}=\frac{\omega}{\kappa}, for the phase velocity, this only makes sense if I use the form (3) for the traveling wave. (the standing wave is not traveling in the x-direction at all, but just going up and down in place; what would be the meaning of a phase velocity in such a situation?)

Furthermore, if I use the formula for group velocity, v_{group}=\frac{d\omega}{d\kappa}, the group velocity is undefined for both the standing wave (2) and the traveling wave (3). (Since κ and ω are both constant, it would not make sense to find the change, dω with respect to a change dκ.)

What kind of wave, y(x,t), yields a well defined group velocity, d\omega/d\kappa, since neither of these waves do?

 

[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.)