Time Dilation & Muon De Broglie Wavelength: Explained

[bobdavis]

I understand it's experimentally verified muons traveling at relativistic speeds relative to an observer will be observed to have longer half-life than would be observed in the rest frame of the muons, which is explained theoretically by a relativistic time dilation effect. Does this correspond to a dilation in the de Broglie wavelength of the muon from the perspective of the observer?

 

[Dale]

Does this correspond to a dilation in the de Broglie wavelength of the muon from the perspective of the observer?

Almost. In relativity the de Broglie relations are given by $$P= \hbar K$$ where in units with $c=1$ we have $P=(E,\vec p)$ and $K=(\omega,\vec k)$

$K$ transforms like a four-vector with $\vec k$ being the spacelike part. So it is more closely related to length contraction than time dilation. And of course the wavelength is the inverse of the wave vector.

 

[bobdavis]

Thanks for the response

I'm not sure I understand yet. Maybe a simpler question, is it the case that all observers would agree on the number of oscillations of the de Broglie wave across the spacetime interval traversed by the muon or is this observer-dependent?

 

[Dale]

is it the case that all observers would agree on the number of oscillations of the de Broglie wave across the spacetime interval traversed by the muon

I don’t know how this is measured or determined, so I cannot say.

 

[PeterDonis]

is it the case that all observers would agree on the number of oscillations of the de Broglie wave across the spacetime interval traversed by the muon or is this observer-dependent?

In any correct relativistic model of a wave, the amplitude and phase of the wave will be invariant functions on spacetime--i.e., at a given spacetime event, the wave will have an amplitude and phase that is independent of any choice of reference frame.

It should be evident that this implies that the number of cycles of any wave along a given curve in spacetime (in this case, a given segment of the worldline of the muon) will be an invariant.

 

[PeterDonis]

I don’t know how this is measured or determined

Mathematically you would look at the 1-form corresponding to the wave vector, and its inner product with the tangent vector to the muon's worldline.

 

[Dale]

Mathematically you would look at the 1-form corresponding to the wave vector, and its inner product with the tangent vector to the muon's worldline.

So $n=K_\mu u^\mu$?

 

[PeterDonis]

So $n=K_\mu u^\mu$?

I think the inner product gives the derivative of $n$, so to speak; the "rate" at which cycles occur along the worldline, i.e., something like $dn / d \tau$. (Just as the wave vector itself gives frequency and wave number, not wavelength and cycle time.)

 

[Orodruin]

I think the inner product gives the derivative of n, so to speak; the "rate" at which cycles occur along the worldline

It does indeed, by definition of the wave 1-form as the exterior derivative of the phase function.

 

[PeroK]

Thanks for the response

I'm not sure I understand yet. Maybe a simpler question, is it the case that all observers would agree on the number of oscillations of the de Broglie wave across the spacetime interval traversed by the muon or is this observer-dependent?

The de Broglie wavelength seems a bit artificial IMO. The wavelength for a massive particle is$$\lambda = \frac h p$$where $p = \gamma mv$ is the (relativistic) momentum.

As a particle has zero momentum in its rest frame, it has infinite de Broglie wavelength in its rest frame. Whether that makes physical sense or not, it doesn't allow an analysis based on simple length contraction.

Note that the de Broglie hypothesis, although still widely taught it seems, was superseded by modern QM in the 1920s.

 

[vanhees71]

In any correct relativistic model of a wave, the amplitude and phase of the wave will be invariant functions on spacetime--i.e., at a given spacetime event, the wave will have an amplitude and phase that is independent of any choice of reference frame.

It should be evident that this implies that the number of cycles of any wave along a given curve in spacetime (in this case, a given segment of the worldline of the muon) will be an invariant.

Just to work this out for an example: Take the electromagnetic field. For simplicity let's consider the four-potential to describe it. For the discussion here gauge dependence does play no significant role anyway. The four-potential transforms as a four-vector field, i.e., under Lorentz transformations
$$A^{\prime \mu}(x')={\Lambda^{\mu}}_{\nu} A^{\nu}(x)={\Lambda^{\mu}}_{\nu} A^{\nu}(\hat{\Lambda}^{-1}x').$$
For a plane wave
$$A^{\mu}(x)=a^{\mu} \exp(\mathrm{i} k \cdot x).$$
Here $a^{\mu}=\text{const}$, and $k=(\omega,\vec{k})$ is the wave four-vector. The above transformation laws tells us that
$$A^{\prime \mu}(x')={\Lambda^{\mu}}_{\nu} a^{\nu} \exp(\mathrm{i} k \cdot \hat{\Lambda}^{-1} x').$$
Since now
$$k \cdot \hat{\Lambda}^{-1} x' = (\hat{\Lambda} k) \cdot (\hat{\Lambda} \hat{\Lambda^{-1}} x') = k' \cdot x',$$
this indeed implies that the phase $k \cdot x$ is indeed an invariant (it's already written as a scalar, and thus that's obvious even without the above derivation). The amplitude $a^{\mu}$ transforms as components of a four-vector and thus the four-vector itself is invariant.

 

[vanhees71]

I think the inner product gives the derivative of $n$, so to speak; the "rate" at which cycles occur along the worldline, i.e., something like $dn / d \tau$. (Just as the wave vector itself gives frequency and wave number, not wavelength and cycle time.)

If $u^{\mu}$ is the four-velocity of an observer of the wave, $u_{\mu} k^{\mu}$ is the angular frequency $\omega=2 \pi f$ of the wave as measured by the observer.

The transformation behavior of the wave four-vector describes the Doppler effect and aberration. For more on relativistic waves, see

https://itp.uni-frankfurt.de/~hees/pf-faq/rela-waves.pdf

 

[Orodruin]

If $u^{\mu}$ is the four-velocity of an observer of the wave, $u_{\mu} k^{\mu}$ is the angular frequency $\omega=2 \pi f$ of the wave as measured by the observer.

The transformation behavior of the wave four-vector describes the Doppler effect and aberration. For more on relativistic waves, see

https://itp.uni-frankfurt.de/~hees/pf-faq/rela-waves.pdf

I’d argue it is even more fundamental. For any function on spacetime or a general manifold, ie, in particular any phase function $\phi$, the one-form $k = d\phi$ describes changes in $\phi$ around each point. With $\dot \gamma$ being the tangent vector of some curve (in particular the world line of an observer in spacetime), the change of $\phi$ with the curve parameter $s$ is given by $d\phi/ds = d\phi(\dot\gamma) = \dot\gamma^\mu \partial_\mu \phi = \dot\gamma^\mu k_\mu$. This is by definition of the exterior derivative.

The phase change per proper time in the case of waves in spacetime is the observed frequency (up to factors of 2 and pi which are only relevant for publication purposes according to Feynman )

 

[vanhees71]

Indeed, this generalizes the "plane-wave solutions" of Minkowski space time to general non-affine spacetimes (like the pseudo-Riemannian spacetime description of GR). It's called the eikonal approximation.

 

[bobdavis]

In any correct relativistic model of a wave, the amplitude and phase of the wave will be invariant functions on spacetime--i.e., at a given spacetime event, the wave will have an amplitude and phase that is independent of any choice of reference frame.

It should be evident that this implies that the number of cycles of any wave along a given curve in spacetime (in this case, a given segment of the worldline of the muon) will be an invariant.

I see, thank you for the response.

It is evident to me that if phase is independent of reference frame then the number of cycles along a given curve will be independent as well.

I have two follow up questions:

1) Have there been any experiments which directly verify this frame-independence of de Broglie wave phase?
2) Does this imply that we can express a particle's half-life in a frame-independent way in terms of the average number of cycles of the de Broglie wave that will occur before particle decay?

 

[Vanadium 50]

I think this is, by its nature, a self-hijacking thread.,

The de Broglie wavelength is more of a step leading to QM than QM itself. I would go further and say it's not much of a theory as it has little predictive power. Once we had the Schroeding Equation, anout a decade later, this idea became obsolete - of historic interest more than physics.

Next, even the Schroeding Equation is non-relativistic. Taking a non-relativistic concept (either Schroeding or de Broglie) and expecting it to make sense relativisticlly (e.g. time dilation) is going to be a lesson in futility.

 

[bobdavis]

I think this is, by its nature, a self-hijacking thread.,

The de Broglie wavelength is more of a step leading to QM than QM itself. I would go further and say it's not much of a theory as it has little predictive power. Once we had the Schroeding Equation, anout a decade later, this idea became obsolete - of historic interest more than physics.

Next, even the Schroeding Equation is non-relativistic. Taking a non-relativistic concept (either Schroeding or de Broglie) and expecting it to make sense relativisticlly (e.g. time dilation) is going to be a lesson in futility.

Is it not ok for me to ask here about what the logical consequences would be of using the de Broglie model and the special relativity model together?

 

[PeroK]

1) Have there been any experiments which directly verify this frame-independence of de Broglie wave phase?
2) Does this imply that we can express a particle's half-life in a frame-independent way in terms of the average number of cycles of the de Broglie wave that will occur before particle decay?

I think you are going astray here. The wavelength of a muon is a theoretical construction that explains diffraction. Likewise its frequency (cycles per second) is not something directly measurable. In the sense that the muon is not oscillating between two states with this frequency.

The observed half-life of the muon can be calculated directly from the proper time that elapses for the muon, which in relativity is the length of its spacetime path.

One of the things that modern QM achieved was to explain experimental wave-particle duality. The de Broglie model in a sense created the wave-particle duality - which QM then explained. It was a big step in the right direction, but you shouldn't take it too literally.

 

[Vanadium 50]

Is it not ok for me to ask here about what the logical consequences would be of using the de Broglie model and the special relativity model together?

Yes, it is not. It would be like trying to understand chemistry using only the elements earth, fire, air and water.

In both cases, we've moved beyond this.

 

[bobdavis]

I think you are going astray here. The wavelength of a muon is a theoretical construction that explains diffraction. Likewise its frequency (cycles per second) is not something directly measurable. In the sense that the muon is not oscillating between two states with this frequency.

The observed half-life of the muon can be calculated directly from the proper time that elapses for the muon, which in relativity is the length of its spacetime path.

One of the things that modern QM achieved was to explain experimental wave-particle duality. The de Broglie model in a sense created the wave-particle duality - which QM then explained. It was a big step in the right direction, but you shouldn't take it too literally.

I see, so if I'm understanding correctly, basically the only measurable prediction we get from the de Broglie model is the diffraction patterns at the detector, and it would be unjustified to extrapolate this to some kind of actual sinusoid wave pattern of the muon during the interval between creation and detection.

So the answer to (1) would be partly yes, in the sense that all observers would predict/measure the same diffraction patterns at the detector, and partly N/A in the sense that the model gives no measurable predictions to verify/refute in the interval between creation & detection anyway; and the answer to (2) would be yes, but only mathematically, and only because using the number of cycles of a theoretical de Broglie wave gives a roundabout way of expressing the length of a spacetime interval, and in that case we could just as well use the length of the spacetime interval directly instead of overcomplicating it with these extra considerations that go beyond the actual applicability of the de Broglie model?

 

[PeroK]

I see, so if I'm understanding correctly, basically the only measurable prediction we get from the de Broglie model is the diffraction patterns at the detector, and it would be unjustified to extrapolate this to some kind of actual sinusoid wave pattern of the muon during the interval between creation and detection.

So the answer to (1) would be partly yes, in the sense that all observers would predict/measure the same diffraction patterns at the detector, and partly N/A in the sense that the model gives no measurable predictions to verify/refute in the interval between creation & detection anyway; and the answer to (2) would be yes, but only mathematically, and only because using the number of cycles of a theoretical de Broglie wave gives a roundabout way of expressing the length of a spacetime interval, and in that case we could just as well use the length of the spacetime interval directly instead of overcomplicating it with these extra considerations that go beyond the actual applicability of the de Broglie model?

I think you're flogging an extinct horse here. The de Broglie model was a transient theory that was a major step on the way to QM; but it's not part of modern QM.

 

[bobdavis]

I'm trying to abide by the standards of the forum but I guess I don't see why my questions are considered invalid and comparable to asking chemistry questions using the 4 element "model". If I understand correctly, unlike the 4 element model of chemistry, the de Broglie model:

1) does make verifiable predictions and has a domain of applicability in which it is valid to use
2) is still used in standard introductory physics education as an educational stepping stone

So to me it's not immediately obvious why I should have thought my questions were more akin to asking about 4 element chemistry than to asking about classical electromagnetism in Newtonian physics, even though the world isn't actually Newtonian and many aspects of electromagnetism are now understood to be purely observer-dependent relativistic effects.

It may very well be the case that there are issues with my questions but just telling me that my questions are so stupid that I shouldn't even have asked them here and saying I should use "modern QM" (which, clearly since I'm asking the questions I'm not quite at that point in my education on quantum physics) doesn't tell me anything that would improve my understanding instead of just shutting down my attempt to inquire, such as: whether I'm extrapolating the model beyond the domain of applicability; whether I'm reasoning incorrectly within the domain of applicability; whether I'm reasoning correctly but incorporating unnecessary theoretical baggage (apparently all 3 of these have been the case?); whether any analogous questions could be asked in the modern QM model of the scenario that might lead to more useful insights or whether the modern QM model is such a departure that there's not even any analogous questions to be asked.

If any of these things were obvious to me ahead of time, I would have asked different questions.

 

[PeroK]

1) does make verifiable predictions and has a domain of applicability in which it is valid to use

You're not staying within its domain of applicability. You want to use it to analyse and understand high-energy muon lifetimes.

 

[Vanadium 50]

You're not staying within its domain of applicability. You want to use it to analyse and understand high-energy muon lifetimes.

Stated another way, he is using a low-velocity approximation to study a high velocity system. And it's not working.

 

[Dale]

Yes, but asking which low velocity laws survive at relativistic speeds is perfectly acceptable. I don’t think we should be discouraging these questions. How is a new student to know which laws are non-relativistic and which are relativistic without asking a question like this?

@bobdavis your questions seem fine to me

 

[Vanadium 50]

The question was asked, and answered. We can't make people accept the answer.

 

[Dale]

this indeed implies that the phase k⋅x is indeed an invariant (it's already written as a scalar, and thus that's obvious even without the above derivation)

My hesitance above was more because I don’t know if the phase of a de Broglie wave is measurable. If not, then it would not need to be invariant. But in any case you are clearly right that since it has a wave four vector it makes sense that the phase should be an invariant also. I didn’t think it through carefully

 

[Mister T]

The deBroglie wave was part of the old quantum theory. It is still used to introduce students to quantum theory, but it has no place in physics and that has been the case for almost a century.

You are confused because a remnant of it remains in physics, and that is simply to refer to the quantity $h/p$ as the deBroglie wavelength.

The upshot of it all is that there is no deBroglie wave, but there is a deBroglie wavelength. It's an unfortunate linguistic remnant.