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Okay, I'll let @ergospherical explain that!Philip Koeck said:In post 24 it says vΦ= 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 said:In post 24 it says vΦ= 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).
Yes, I'm getting that impression.ergospherical said:Particles are not waves and you should not confuse their equations - phase velocity isn't so meaningful for particles.
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.Philip Koeck said:At least to me that leads to a problem.
In relativity rest mass is part of the total energy. So it has to be included.Philip Koeck said:I'm simply wondering which of the two is correct for relativistic velocities.
Could you share a source for the relationships you use in the first line?ergospherical said: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)##.
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.Philip Koeck said:When I write m I mean the relativistic mass, not the rest mass.