Phase velocity and frequency of a matter wave

[PeroK]

In post 24 it says v_Φ= ω/k, and then ω is replaced by E - mc², as I read it. (The h cancels or is set to 1, c is set to 1).

Okay, I'll let @ergospherical explain that!

 

[Philip Koeck]

Particles are not waves and you should not confuse their equations - phase velocity isn't so meaningful for particles.

Yes, I'm getting that impression.
Just to make sure, from your expression for v_Φ, what do you get when v approaches c and does v_Φ ever exceed c (as the Wikipedia text claims)?

 

[PeterDonis]

At least to me that leads to a problem.

If it's enough of a problem that its effect on your answer matters, then obviously you are dealing with a case where the non-relativistic approximation won't work for you and you need to use the relativistic equations.

 

[PeterDonis]

I'm simply wondering which of the two is correct for relativistic velocities.

In relativity rest mass is part of the total energy. So it has to be included.

 

[Philip Koeck]

In the relativistic region the phase velocity is $v_{\phi} = \omega/k = (E-m)/p$, i.e.\begin{align*}
v_{\phi} = \frac{\gamma - 1}{\gamma v}
\end{align*}For small $v$ expand $\gamma = (1-v^2)^{-1/2} \sim 1 + v^2/2$ (and similarly $1/\gamma \sim 1-v^2/2$) to get\begin{align*}
v_{\phi} &\sim \frac{(v^2/2)}{v}(1-v^2/2) \sim \frac{v}{2}
\end{align*}omitting terms $O(v^3)$.

Could you share a source for the relationships you use in the first line?
Especially E -mc²= ħω, if that is what you are using.
Is there a book that says that?

 

[Nugatory]

When I write m I mean the relativistic mass, not the rest mass.

Don't do that. It kinda sort of works if you're just doing a semester or so of special relativity with classical objects (but even that is discouraged these days - why start with something that has to be unlearned to move beyond that first semester or so?) but fails dismally as soon as you start thinking quantum mechanics.

With QM in the picture: If relativity doesn't matter there is no relativistic mass, if it does matter you have to use the relativistic $E^2=(mc^2)^2+(pc)^2$ instead of anything involvng relativistic mass.

 

[Philip Koeck]

A (hopefully) final question on this:
Clearly the wavelength of a matter wave can be measured directly in a diffraction experiment so the relation p = h k can be tested experimentally by sending particles of known p through a grating.

Is there a similar way to test E = h f?
Can f be measured directly without inferring it from E?

Or, alternatively, is there a way to measure the phase velocity of a matter wave?